On the Method of Correspondent Values, &c. By Edward Waring, M. D. F. R. S. and Lucasian Professor of the Mathematics at Cambridge

XVII. On the Method of correspondent Values, &c. By Edward Waring, M.D. F.R.S. and Lucasian Professor of the Mathematics at Cambridge. Read May 28, 1789. 1. In the year 1762 I published a method of finding when two roots of a given equation $x^n - px^{n-1} + qx^{n-2} - rx^{n-3} + \ldots$ are equal, by finding the common divisors of the two quantities $a^n - pa^{n-1} + qa^{n-2} - \ldots$, and $na^{n-1} - n-1pa^{n-2} + n-2qa^{n-3} - \ldots$, and observed if they admitted only one simple divisor $(a-A)$, then two roots were only equal; if a quadratic $(a^2 - Aa + B)$, then two roots of the equation became twice equal; if a cubic $(a^3 - Aa^2 + Ba - C)$, then two roots became thrice equal; and so on: or, to express in more general terms what follows from the same principles, if the common divisor be $a-b' \times a-c' \times a-d' \times \ldots$, then $r+1$ roots of the given equation will be $b$, $s+1$ roots will be $c$, $t+1$ will be $d$, &c.; and it immediately follows, from the principles delivered in the second edition of the same Book, published in 1770, that to find when $r+1$, $v+1$, $t+1$, &c. roots are respectively equal requires $r+s+t$, &c. equations of condition, which are deducible from the well known method of finding the common divisors of two quantities in this case of $a^n - pa^{n-1} + qa^{n-2} - \ldots$, $na^{n-1} - n-1pa^{n-2} + n-2qa^{n-3} - \ldots$ of the terms of their remainders, &c. In the book above mentioned the equations of condition are given, which discover when two roots are equal in the equations \(x^3 - px^2 + qx - r = 0\), \(x^4 + qx^2 - rx + s = 0\), \(x^5 + qx^3 - rx^2 + sx - t = 0\), in the two latter equations the second term is wanting, which may easily be exterminated; but it may as easily be restored by substituting for \(q\), \(r\), \(s\), &c. in the equation of condition found the quantities resulting from the common transformation of equations to destroy the second term. 2. Another rule contained in the same Book is the substitution of the roots of the equation \(na^{n-1} - n-1pa^{n-2} + n-2qa^{n-3} - \&c.\) respectively for \(a\) in the quantity \(a^n - pa^{n-1} + qa^{n-2} - \&c.\), and multiplication of all the quantities resulting into each other; their content will give the equation of condition, when two roots are equal. Mr. Huddes first discovered, that if the successive terms of the given equation are multiplied into an arithmetical series, the resulting equation will contain one of any two equal roots, and \(m\) of the \(m+1\) equal roots in the given equation. 3. If \(3, 4, 5, \ldots r\) roots of the equation are equal, find a common divisor of \(3, 4, 5, \ldots r\) of the subsequent quantities \(a^n - pa^{n-1} + qa^{n-2} - \&c.\), \(na^{n-1} - n-1pa^{n-2} + n-2qa^{n-3} - \&c.\), \(n \cdot n-1a^{n-2} - n-1 \cdot n-2pa^{n-3} + n-2 \cdot n-3qa^{n-4} - n-3 \cdot n-4ra^{n-5} + \&c.\), \(n \cdot n-1 \cdot n-2a^{n-3} - n-1 \cdot n-2 \cdot n-3pa^{n-4} + n-2 \cdot n-3 \cdot n-4qa^{n-5} - \&c., \ldots n \cdot n-1 \cdot n-2 \cdot n-r+2a^{n-r+1} - n-1 \cdot n-2 \cdot n-r+1pa^{n-r} + \&c.;\) which will probably be best done by dividing all the preceding quantities by the quantity of the least dimension of \(a\), and the divisor and all the remainders by that quantity which has the least dimensions amongst them; and so on: there will result \(2, 3, 4, \ldots r-1\) equations of condition; and in this case it is observed, Dr. Waring on the observed, in the before-mentioned Book, that (if the common divisor be \((a - A)\)) it will once only admit of \(3, 4, 5, \ldots r\) equal roots; if it be a quadratic, then it will twice admit of those equal roots; and so on. 4. If the roots of the equation of the least dimensions be substituted for \(a\) in the remaining equations, and each of the resulting values of the same equation be multiplied into each other, there will result the \(r - 1\) equations of condition: and the same may be deduced also from the several equations conjointly. The equations of conditions found by the first method, if the divisions were not properly instituted, may admit of more rational divisors than necessary, of which some are the equations of conditions required. 2. 1. In the year 1776, I published in the Meditationes Analyticæ a new method of differences for the resolution of the following problem. Given the sums of a swiftly converging series \(ax + bx^2 + cx^3 + dx^4 + \&c.\), when the values of \(x\) are respectively \(\pi, \rho, \sigma, \&c.\); to find the sum of the series when \(x\) is \(\tau\), that is, given \(S_\pi = a\pi + b\pi^2 + c\pi^3 + d\pi^4 + \&c., S_\rho = a\rho + b\rho^2 + c\rho^3 + \&c., S_\sigma = a\sigma + b\sigma^2 + c\sigma^3 + \&c. \&c.;\) to find \(S_\tau = a\tau + b\tau^2 + c\tau^3 + \&c.\). To resolve this problem I multiplied the quantities, \(S_\pi, S_\rho, S_\sigma, \&c.\) respectively into unknown co-efficients \(\alpha, \beta, \gamma, \&c.\) and there resulted. Method of correspondent Values, &c. \[ \alpha \pi a + \alpha \pi^2 b + \alpha \pi^3 c + \&c. \] \[ \beta \rho a + \beta \rho^2 b + \beta \rho^3 c + \&c. \] \[ \gamma \sigma a + \gamma \sigma^2 b + \gamma \sigma^3 c + \&c. \] \&c. \&c. \&c. and then made the sum of each of the terms respectively equal to its correspondent term of the quantity \( r a + r^2 b + r^3 c + \&c. \), and consequently \( \alpha \pi + \beta \rho + \gamma \sigma + \&c. = r, \alpha \pi^2 + \beta \rho^2 + \gamma \sigma^2 + \&c. = r^2, \alpha \pi^3 + \beta \rho^3 + \gamma \sigma^3 + \&c. = r^3, \&c. \) I assumed as many equations of this kind as there were given values \( \pi, \rho, \sigma, \&c. \) of \( x \); and consequently as many equations resulted as unknown quantities \( \alpha, \beta, \gamma, \&c.; \) whence, by the common resolution of simple equations, or more easily from differences, can be found the unknown quantities \( \alpha, \beta, \gamma, \&c., \) and thence the equation sought \( \alpha \times S \pi + \beta \times S \rho + \gamma \times S \sigma + \&c. = S r \) nearly. 3. In the Meditationes are assumed for \( \pi, \rho, \sigma, \&c. \) the quantities \( p, 2p, 3p, 4p, \ldots n - 2p, n - 1p, \) and \( np \) for \( r; \) which, if substituted for their values in the preceding equations, will give \( \alpha + 2\beta + 3\gamma + 4\delta + \&c. = n, \alpha + 4\beta + 9\gamma + 16\delta + \&c. = n^2, \alpha + 8\beta + 27\gamma + \&c. = n^3, \alpha + 16\beta + 81\gamma + \&c. = n^4; \) and if the sums of the series \( ax + bx^2 + cx^3 + \&c. \) which respectively correspond to the values \( p, 2p, 3p, \ldots n - 1p \) of \( x \) be \( S_1, S_2, S_3, S_4, \ldots S_{n-1}, \) and the sum of the series \( ax + bx^2 + cx^3 + \&c. \) which corresponds to \( n \) value of \( x \) be \( S_n; \) then will \[ S_n = nS_{n-1} - n \cdot \frac{n-1}{2} S_{n-2} + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{2} S_{n-3} \ldots = nS_1 \] nearly, which equation is given in the above-mentioned Book. 3. The logarithm from the number, the arc from the sine, &c. are found by serieses of the formula \( ax + bx^2 + cx^3 + \&c.; \) and consequently this equation is applicable to them. 6. Inv. 4. In the same Book is assumed a series $ax^r + bx^{r+s} + cx^{r+2s} + dx^{r+3s} + \&c.$ of a more general formula than the preceding, and in it for $x$ substituted $\alpha, \beta, \gamma, \delta, \&c., m$; and $S\alpha, S\beta, S\gamma, S\delta, \&c.; Sm$ for the resulting sums, and thence deduced $$Sm = \frac{m^r \times m^i - \beta^i \cdot m^i - \gamma^i \cdot m^i - \delta^i \cdot \&c.}{a^r \times a^i - \beta^i \cdot a^i - \gamma^i \cdot a^i - \delta^i \cdot \&c.} \times S\alpha + \frac{m^r \times m^i - \alpha^i \cdot m^i - \gamma^i \cdot m^i - \delta^i \cdot \&c.}{\beta^r \times \beta^i - \alpha^i \cdot \beta^i - \gamma^i \cdot \beta^i - \delta^i \cdot \&c.} \times S\beta + \frac{m^r \times m^i - \alpha^i \cdot m^i - \beta^i \cdot m^i - \delta^i \cdot \&c.}{\gamma^r \times \gamma^i - \alpha^i \cdot \gamma^i - \beta^i \cdot \gamma^i - \delta^i \cdot \&c.} \times S\gamma + \frac{m^r \times m^i - \alpha^i \cdot m^i - \beta^i \cdot m^i - \gamma^i \cdot \&c.}{\delta^r \times \delta^i - \alpha^i \cdot \delta^i - \beta^i \cdot \delta^i - \gamma^i \cdot \&c.} \times S\delta + \&c.$$ nearly. Cor. If for $r$ and $s$ be assumed respectively 1, the series becomes $ax + bx^2 + cx^3 + \&c.$ of the same formula as the preceding: if $r = 0$ and $s = 1$, the series becomes $a + bx + cx^2 + \&c.$ The latter case will be the same as the former, when one of the quantities ($\alpha$) substituted for $x$ and its correspondent sum $S\alpha$, both become = 0, and the equation deduced in both cases the same. 5. If $\pi, \rho, \sigma, \&c.$ respectively denote $r, r+p, r+2p, \&c.$ $r+n-2p, r+n-1p,$ and $\tau = r+np$; and $S, S_1, S_2, S_3, \&c.$ $Sn-2, Sn-1$, be the sums either resulting from the series $ax + bx^2 + cx^3 + \&c.$ or the series $A + ax + bx^2 + cx^3 + \&c.$, which respectively correspond to the values $r, r+p, r+2p, \&c.$ of $x$; and $Sn$ the sum of the same series which corresponds to the value $r+np$ of $x$; then will $Sn = nSn-1 - n \cdot \frac{n-1}{2}Sn-2 + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3}Sn-3 - \&c.$ $= n \cdot \frac{n-1}{2}S_2 = nS_1 = S$ nearly; this equation differs from the preceding by the last term $S$ not vanishing; in the preceding case $S$ became = 0, for it was the sum of the series $ax + bx^2 + cx^3 + \&c.$, which corresponded to $x = 0$. 6. From 6. From the Meditationes it appears, that \( r^m - n \times r = p^m + n \times \frac{n-1}{2} r = 2p^m - n \times \frac{n-1}{2} \times \frac{n-2}{3} r = 3p^m + \&c. \) to the end of the series \( = 0 \), if \( m \) is less than \( n \), and \( m \) and \( n \) are whole numbers; but if \( m = n \), then it will \( = \pm 1 \cdot 2 \cdot 3 \cdot 4 \cdots \frac{n-1}{n} np^m \); whence it is manifest, that for the \( n \) first terms of the series \( A + ax + bx^2 + \&c. \) the equations are true; and for the \( n - 1 \) first terms of the series \( ax + bx^2 + cx^3 + \&c. \) and in the successive term of both the series they will err by a quantity nearly \( = \pm 1 \cdot 2 \cdot 3 \cdots n \times p^n \times r^{-n} \times \text{co-efficient of the term}; \) and the errors of every subsequent term \( (x^{4+n}) \) will be nearly as \( \pm m \cdot \frac{m-1}{2} \cdot \frac{m-2}{3} \cdot \frac{m-3}{4} \cdots \frac{m-b+1}{b} \times p^n \times r^{-n} \times \text{co-efficient of the term} x^{b+n}, \) if for \( r, r + p, r + 2p, \&c. \) be substituted \( 1, 1 + \frac{p}{r}, 1 + \frac{2p}{r}, \&c. \) 7. Let the preceding equation \( S_n = nS_{n-1} - n \cdot \frac{n-1}{2} S_{n-2} + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot S_{n-3} - \&c. = n \times \log. r - p - n \cdot \frac{n-1}{2} \log. r - 2p + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \log. r - 3p + \&c. = \log. \frac{r \times r - 2p' \times r - 4p' \times r - 6p' \times \&c.}{r - p' \times r - 3p' \times r - 5p' \times \&c.} = \log. K, \) where \( s, s', s'', \&c. \) denote the co-efficients of the alternate terms of the binomial theorem, viz. \( s = n \cdot \frac{n-1}{2}, s' = n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot \frac{n-3}{4}, \&c., \) and \( t = n, t' = n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3}, \&c. \) the co-efficients of the remaining alternate terms; the numerator \( r \times r - 2p' \times r - 4p' \times r - 6p' \times \&c. = (if N = 2^{n-1}) r^N - Ppr^{N-1} + Qp^er^{N-2} - Rp^3r^{N-3} \cdots Lp^{n-1} \times r^{N-n+1} = Mp^n r^{N-n} + \&c.; \) and the denominator \( \frac{r - p' \times r - 3p' \times r - 5p' \times \&c.}{1} = r^N - Ppr^{N-1} + Qp^2r^{N-2} - Rp^3r^{N-3}. \) Dr. Waring on the \[ R p^3 r^{n-3} + \ldots L p^{n-r} N^{-n+1} (M + 1 \cdot 2 \cdot 3 \ldots n - 1) p^n r^{N-n} = \ldots \] &c., whence the numerator and denominator have the \( n \) first terms the same, and the next succeeding terms differ by \[ 1 \cdot 2 \cdot 3 \ldots n - 1 p^n r^{N-n} \] the numerator divided by the denominator \( = \frac{1 \cdot 2 \cdot 3 \ldots n - 1}{r^n} p^n \) nearly, if \( r \) be a great number in proportion to \( p \), &c. it would be \( + \) when \( n \) is an odd number, and \( - \) when even. 8. The logarithm of the fraction \( K \) by the common series \[ = K - 1 - \frac{K - 1}{2} + \frac{K - 1}{3} - \ldots \] has for its first term \( = \frac{1 \cdot 2 \cdot 3 \ldots n - 1}{r^n} \times p^n \) nearly; for its second term the square of the first divided by 2, &c. 9. The error of this equation not only depends on the logarithm of \( K \), which may be calculated to any degree of exactness, but in the calculus on the errors of the given logarithms. 10. If \( r \) be increased or diminished by any given number, the \( n \) first terms of the numerator and denominator will still result the same, and the next succeeding terms will differ by \[ 1 \cdot 2 \cdot 3 \cdot 4 \ldots n - 1 \times p^n \times r^{N-n}. \] 11. Let \( n \cdot \frac{n-1}{2} \) numbers be 2, \( n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot \frac{n-3}{4} \) numbers be 4, \( n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot \frac{n-3}{4} \cdot \frac{n-4}{5} \cdot \frac{n-5}{6} \) numbers be 6, &c.; their sum, the sum of the products of every two, the contents of every three, four, five, &c. to \( n - 1 \) of them will be equal to the sum, the sum of the products of every two, of the contents of every three, four, five, &c. to \( n - 1 \) of the following numbers, viz. \( n \) numbers which are 1, \( n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \) numbers numbers which are $3, \frac{n-1}{2}, \frac{n-2}{3}, \frac{n-3}{4}, \frac{n-4}{5}$, which are $5$, &c.; and the sum of the contents of every $n$ of the former will be less than the sum of the contents of every $n$ latter numbers by $1 \cdot 2 \cdot 3 \cdot 4 \ldots n - 1$. 12. The method given in Art. 4. which I name a method of correspondent values, easily deduces and demonstrates the pre- ceding equations, which cannot, without much difficulty, be done by the preceding method of differences; the method of correspondent values is much preferable to the method of dif- ferences, both for the facility of its deduction, and the gene- rality of its resolution: for instance, from this method very easily can be deduced, &c. the subsequent and other similar equations. Ex. 1. $S_n = nS_{n-1} - n \cdot \frac{n-1}{2} S_{n-2} + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} S_{n-3} \ldots$ $\equiv nS_1 \equiv S$ nearly. Ex. 2. $S_n + m = \frac{m+n}{1} \cdot \frac{m+n-1}{3} \cdot \frac{m+n-2}{n-1} \ldots \frac{m+2}{n-1} \times S_{n-1} - \frac{n-1}{1} \times A \times \frac{m+1}{m+2} \times S_{n-2} + \frac{n-2}{2} \times B \times \frac{m+2}{m+3} \times S_{n-3} - \frac{n-3}{3} \times C \times \frac{m+3}{m+4} \times S_{n-4} + \frac{n-4}{4} \times D \times \frac{m+4}{m+5} \times S_{n-5} - \&c.$ nearly, where the letters $A, B, C, D, \&c.$ denote the preceding co-efficients, and the converging series is the same as in the preceding example. Ex. 3. Let the converging series be of the formula $ax + bx^2$ $cx^5 + dx^7 + \&c.$; then will $S_n = 2n-2 S_{n-1} - 2n-1 \times \frac{2n-4}{2} S_{n-2} + 2n-1 \times \frac{2n-2}{2} \times \frac{2n-6}{3} S_{n-3} - 2n-1 \cdot \frac{2n-2}{2} \cdot \frac{2n-3}{2} \times \frac{2n-8}{4} S_{n-4} + \&c.$ nearly, of which the general term is $\frac{2n-1}{2} \cdot \frac{2n-2}{2} \cdot \frac{2n-3}{3} \ldots \frac{2n-l+1}{l-1} \times \frac{2n-2l}{l} \times S_{n-l}$. Ex. 4. Let the series be of the formula $A + ax^2 + bx^4 + cx^6 + \&c.$; then will $S_n = \frac{n-1}{n} \times \frac{2n-2}{n} S_{n-1} - \frac{n-2}{n} \times \frac{2n-1}{n} \cdot \frac{2n-4}{2} \times S_{n-2} + \frac{n-3}{n} \times \frac{2n-1}{n} \cdot \frac{2n-2}{2} \cdot \frac{2n-6}{3} S_{n-3} - \frac{n-4}{n} \times \frac{2n-1}{n} \cdot \frac{2n-2}{2} \cdot \frac{2n-3}{3} \times \frac{2n-8}{4} S_{n-4} + \&c.$ nearly, of which the general term is $\frac{n-1}{n} \times \frac{2n-1}{n} \cdot \frac{2n-2}{3} \cdot \frac{2n-3}{3} \cdots \frac{2n-l+1}{l-1} \times \frac{2n-2l}{l} \times S_{n-l}$. Ex. 5. Let the given series be of the formula $ax + bx^2 + cx^3 + \&c.$, and in it for $x$ be substituted $p$, $-p$, $2p$, $-2p$, $3p$, $-3p$, $\ldots$, $np$, $-np$ and $mp$, and for the sums of the resulting serieses be wrote respectively $S^1$, $S^{-1}$, $S^2$, $S^{-2}$, $S^3$, $S^{-3}$, $\ldots$, $S^n$, $S^{-n}$, and $Sm$; then will $Sm =$ $$\frac{m \cdot m^2 - 1 \cdot m^2 - 4 \cdot m^2 - 9 \cdot m^2 - 16 \cdots m^2 - n - 1}{n \cdot n^2 - 1 \cdot n^2 - 4 \cdot n^2 - 9 \cdot n^2 - 16 \cdots n^2 - n - 1} \times 2n = 1 \cdot 2 \cdot 3 \cdot 4 \cdots 2n$$ $$\times S^{-n} + A \times \frac{m+n}{m-n} S^{n} - \frac{2n}{1} \times B \times \frac{m-n}{m+n-1} S^{-n+1} - C \times \frac{m+n-1}{m-n-1} \times$$ $$S^{n-1} + \frac{2n-1}{2} \times D \times \frac{m-n-1}{m+n-2} \times S^{-n+2} + \frac{m+n-2}{m-n-2} \times E S^{n-2} - \frac{2n-2}{3} \times$$ $$F \times \frac{m-n-2}{m+n-3} S^{-n+3} - G \times \frac{m+n-3}{m-n-3} \times S^{n-3} + \&c.$$ nearly, where the letters $A$, $B$, $C$, $D$, $\&c.$ respectively denote the preceding co-efficients. In general, the co-efficients of the terms $S^{-n+i}$ and $S^{-n-i}$ will be respectively $M = \frac{2n-s+1}{s} \times L \times \frac{m-n-s+1}{m+n-s}$ and $M \times \frac{m+n-s}{m-n-s}$, where the letters $L$ and $M$ respectively denote their preceding co-efficients; the co-efficients are to be taken affirmatively, or negatively, according as $s$ is an even or odd number. Ex. 6. If for $x$ in the preceding series be substituted $p$, $-p$, $2p$, $-2p$, $3p$, $-3p$, $\ldots$, $n-1p$, $-n-1p$, $np$ respectively, then $Sm$. Method of correspondent Values, &c. \[ S_m = \frac{m \cdot m^2 - 1 \cdot m^2 - 4 \cdot m^2 - 9 \cdot m^2 - 16 \cdots (m^2 - (n-1)^2)}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdots (2n-1)} \] \[ S^n = A \times \frac{m-n}{m+n-1} \times S^{-n+1} - \frac{2n-1}{1} \times B \times \frac{m+n-1}{m-n-1} \times S^{-n} + C \times \frac{m-n-1}{m+n-2} \times S^{-n+2} + \frac{2n-2}{2} \times D \times \frac{m+n-2}{m-n-2} \times S^{-n-2} - \&c. \text{ nearly, when } A, B, C, \&c. \text{ denote as before the preceding co-efficients. The coefficients of the terms } S^{-n+1} \text{ and } S^{-n} \text{ will be respectively } L \times \frac{m-n-s+1}{m+n-s} \text{ and } \frac{2n-1}{s} \times M \times \frac{m+n-s}{m-n-s}, \text{ L and M denoting the preceding co-efficients, which are to be taken negatively or affirmatively, as } s \text{ is an even or an odd number. In this series when } x = 0, \text{ the correspondent sum } = 0. \] Ex. 7. Let the given series be of the formula \( a + bx + cx^2 + dx^3 + \&c.; \) and in it for \( x \) be substituted \( o, p, -p, 2p, -2p, 3p, -3p \cdots np, -pp \text{ and } mp, \) and for the sums of the resulting serieses be wrote as before \( S^o, S^1, S^{-1}, S^2, S^{-2}, \ldots S^n, S^{-n}, \text{ and } S^m; \) then will \[ S_m = \frac{m \cdot m^2 - 1 \cdot m^2 - 4 \cdot m^2 - 9 \cdots (m^2 - n^2)}{n \cdot n^2 - 1 \cdot n^2 - 4 \cdot n^2 - 9 \cdots (n^2 - n^2)} \times \frac{m-n}{2n} \times S^{-n} + A \times \frac{m+n}{m-n} \times S^{+n} - \frac{2n}{1} \times B \times \frac{m-n}{m+n-1} \times S^{-n+1} - \&c. \text{ this series observes the same law as the series given in Ex. 5. and only differs from it by the last term } So \text{ not vanishing, that is, being } = 0. \] Ex. 8. Let the series be of the preceding formula \( a + bx + cx^2 + dx^3 + \&c., \) and in it for \( x \) be substituted \( o; p, -p; 2p, -2p; 3p, -3p; \cdots n-1p, -n-1p, np, \text{ and } mp, \) and the sums resulting be \( So, S_1, S^{-1}, S^2, S^{-2}, \ldots S^{n-1}, S^{-n+1}, S^n \text{ and } S^m; \) then will \[ S_m = \frac{m \cdot m^2 - 1 \cdot m^2 - 4 \cdots m^2 - (n-1)^2}{1 \cdot 2 \cdot 3 \cdot 4 \cdots 2n-1} \times S^n - A \times \frac{m-n}{m+n-1} \times S^{-n+1} - \&c. \text{ the same series as in Ex. 6. and differs from it only by the last term } So \text{ not vanishing. } \] Ex.9. Let the series be of the same formula \(a + bx + cx^2 + dx^3 + \ldots\) and in it for \(x\) be substituted \(p, -p, 3p, -3p, 5p, -5p, 7p, \ldots np, -np\) and \(mp\); and the sums resulting be \(S_1, S_{-1}, S_3, S_{-3}, S_5, S_{-5}, S_7, S_{-7}, \ldots S_n, S_{-n}\), and \(S_m\); then will \[ S_m = \frac{m^2 - 1}{n^2 - 1} \cdot \frac{m^2 - 9}{n^2 - 9} \cdot \frac{m^2 - 25}{n^2 - 25} \cdot \frac{m^2 - 49}{n^2 - 49} \cdots \frac{m^2 - n^2}{n^2 - n^2} \times m + n \] \[ \times S_n - A \times \frac{n - 1}{m + n} \times S_{-n} - \frac{n}{m - n + 2} \times B \times \frac{m + n}{m - n + 2} \times S_{-n} + C \times \frac{m - n + 2}{m + n - 2} \times S_{-n + 2} + \frac{n - 1}{2} \times D \times \frac{m + n - 2}{m - n + 4} \times S_{-n - 4} - E \times \frac{m - n + 4}{m + n - 4} \times S_{-n + 4} - \frac{n - 2}{3} F \] \[ \times \frac{m + n - 4}{m - n + 6} \times S_{-n - 6} + \frac{m - n + 6}{m + n - 6} \times G \times S_{-n + 6} + \frac{n - 3}{4} \times H \times \frac{m + n - 6}{m - n + 8} \times S_{-n - 8} - \ldots \text{&c. nearly, where the letters } A, B, C, D, E, \text{ &c. denote the preceding co-efficients of the terms } S_n, S_{-n}, S_{-n - 2}, S_{-n + 2}, S_{-n - 4}, S_{-n + 4}, S_{-n - 6}, \text{ &c. respectively. The co-efficients of the terms } S_{2n - 2s} \text{ and } S_{-n + 2s} \text{ will be } M = \frac{n - s + 1}{s} \times L \times \frac{m + n - 2s + 2}{m - n + 2s} \text{ and } N = M \times \frac{m - n + 2s}{m + n - 2s}; \text{ where } L, M, \text{ and } N \text{ denote the co-efficients of the terms immediately preceding each other, that is, of the terms } S_{-n + 2s - 2}, S_{-n - 2s}, \text{ and } S_{-n + 2s}. \text{ The sign of the first co-efficient } M \text{ will be } + \text{ or } -, \text{ according as } s \text{ is even or odd; the second term } N \text{ will have a contrary sign to the first.} These series may be made to begin from any term, which may be easily found by the method of correspondent values, and the subsequent terms from it by the given law; its preceding terms may be deduced from the same law reversed, that is, by putting the numerators of the fractions multiplied into it for the denominators, and the denominators for the numerators. From Method of correspondent Values, &c. From these different serieses may be formed, by adding two or more terms of the given series together for a term of the required series; which method has been applied to converging series in general in the Meditationes. 13. The method of correspondent values easily affords a resolution of the problems contained in Mr. Brigg's or Sir Isaac Newton's method of differences. Ex. 1. Let the quantity be of the formula $a + bx + cx^2 + dx^3 + \&c. \ldots x^n = y$, and $n + 1$ correspondent values of $x$ and $y$ be given, viz. $p, q, r, s, \&c.$ of $x$; $Sp, Sq, Sr, Ss, \&c.$ of $y$; then will $$y = \frac{x-q}{p-q} \cdot \frac{x-r}{p-r} \cdot \frac{x-s}{p-s} \cdot \&c. \times Sp + \frac{x-p}{q-p} \cdot \frac{x-r}{q-r} \cdot \frac{x-s}{q-s} \cdot \&c. \times Sq +$$ $$\frac{x-p}{r-p} \cdot \frac{x-q}{r-q} \cdot \frac{x-s}{r-s} \cdot \&c. \times Sr + \frac{x-p}{s-p} \cdot \frac{x-q}{s-q} \cdot \frac{x-s}{s-s} \cdot \&c. \times Ss + \&c.$$ The truth of this problem very easily appears by writing $p, q, r, s, \&c.$ for $x$ in the given series. All the preceding examples may be applied to this case, by writing $x$ for $m$ in the given series; hence the resolutions of several cases of equi-distant ordinates by easy and not inelegant serieses, amongst which are included the two cases commonly given on this subject. 14. If a quantity be required, which proceeds according to the dimensions of $x$, reduce the above given value of $y$ into a quantity proceeding according to the dimensions of $x$, and there results $$y = \left( \frac{Sp}{p-q} \cdot \frac{Sr}{p-r} \cdot \frac{Ss}{p-s} \cdot \&c. = A + \frac{Sq}{q-p} \cdot \frac{Sr}{q-r} \cdot \frac{Ss}{q-s} \cdot \&c. = B + \frac{Sr}{r-p} \cdot \frac{Sr}{r-q} \cdot \frac{Sr}{r-s} \cdot \&c. = C + \frac{Ss}{s-p} \cdot \frac{Ss}{s-q} \cdot \frac{Ss}{s-s} \cdot \&c. = D \right) \times x^n -$$ $$\left( \frac{Sp \times q + r + s + \&c.}{A} + \frac{Sq \times p + r + s + \&c.}{B} + \frac{Sr \times p + q + s + \&c.}{C} + \frac{Ss \times p + q + r + \&c.}{D} + \&c. \right) x^{n-1} + \left( \frac{Sp \times qr + qs + rs + \&c.}{A} + \&c. \right) x^{n-2} + \&c.$$ The law and continuation of this series is evident to any one versant in these matters from inspection. These fractions may be reduced to a common denominator by substituting for $Sp$ and $A$ the products $Sp \times P$ and $A \times P$, where $P = q - r \cdot q - s \cdot r - s \cdot &c.$; for $Sq$ and $B$ the products $Sq \times Q$ and $B \times Q$, where $Q = p - r \cdot p - s \cdot r - s \cdot &c.$; for $Sr$ and $C$ the products $Sr \times R$ and $C \times R$, where $R = p - q \cdot p - s \cdot q - s \cdot &c.$; for $Ss$ and $D$ the products $Ss \times S'$ and $C \times S'$, where $S' = p - q \cdot p - r \cdot q - r \cdot &c.$ &c. The fractions, in particular cases, will often be reducible to lower terms. 15. Let $y = ax^b + bx^{b+1} + cx^{b+2} + &c.$, and the correspondent values of $x$ and $y$ be given as before, then will $y =$ $$\frac{x^b}{p^b} \times \frac{x^i - q^i}{p^i - q^i} \times \frac{x^i - s^i}{p^i - s^i} \times &c. \times Sp + \frac{x^b}{q^b} \times \frac{x^i - p^i}{q^i - p^i} \times \frac{x^i - s^i}{q^i - s^i} \times Sq$$ $$+ \frac{x^b}{r^b} \times \frac{x^i - p^i}{r^i - p^i} \times \frac{x^i - q^i}{r^i - q^i} \times &c. \times Sr + \frac{x^b}{s^b} \times \frac{x^i - p^i}{s^i - p^i} \times \frac{x^i - q^i}{s^i - q^i} \times &c. \times Ss + &c.$$ This series may in the same manner as the preceding be reduced to terms, proceeding according to the dimensions of $x$; and the serieses given in the examples may (mutatis mutandis) be predicated of it. 16. A more general method of correspondent values is given in the Meditationes, as also the subsequently $y =$ $$\frac{x - q}{p - q} \cdot \frac{x - s}{p - s} \cdot &c.$$ Method of correspondent Values, &c. \[ x \cdot Sp + \frac{x-p}{q-p} \cdot \frac{x-r}{q-r} \cdot \frac{x-s}{q-s} \cdot \ldots \cdot Sq + \frac{x-p}{r-p} \cdot \frac{x-q}{r-q} \cdot \frac{x-s}{r-s} \cdot \ldots \cdot Sr + \ldots \] &c. as in Ex. 1. = \( Sp + (x-p) \left( \frac{1}{p-q} \times Sp + \frac{1}{q-p} \times Sq \right) + (x-p) \) \[(x-q) \left( \frac{1}{p-q} \times \frac{1}{p-r} \times Sp + \frac{1}{q-p} \times \frac{1}{q-r} \times Sq + \frac{1}{r-p} \times \frac{1}{r-q} \times Sr \right) + \] \[(x-p)(x-q)(x-r) \left( \frac{1}{p-q} \cdot \frac{1}{p-r} \cdot \frac{1}{p-s} \cdot \ldots \cdot Sp + \frac{1}{q-p} \cdot \frac{1}{q-r} \cdot \frac{1}{q-s} \cdot \ldots \cdot Sq + \frac{1}{r-p} \cdot \frac{1}{r-q} \cdot \frac{1}{r-s} \cdot \ldots \cdot Sr \right) - \ldots \] The equality of these two different quantities will easily appear by finding the co-efficients of both, which are multiplied into the same given value of \( y \) as \( Sp, Sq, Sr, \ldots \), and the same power of \( x \); for with very little difficulty they will in general be found equal. It is evident from this resolution that, giving the ordinates and their respective distances from each other, the value of any other ordinate at a given distance from the preceding, found by this method, will result the same, whatever may be the point assumed from which the absciss is made to begin. 3. 1. Let a series be \( Ax + Bx^2 + Cx^3 + Dx^4 + \ldots \) of such a formula that if in it for \( x \) be substituted \( a + b \), there results a series \[ A \times a + b + B \times a + b^2 + C \times a + b^3 + D \times a + b^4 + \ldots = (Aa + Ba^2 + Ca^3 + Da^4 + \ldots) \times (1 + qb + rb^2 + sb^3 + tb^4 + \ldots) + \] \[(1 + qa + ra^2 + sa^3 + ta^4 + \ldots) \times (Ab + Bb^2 + Cb^3 + Db^4 + \ldots) \] then will the series \( Ax + Bx^2 + Cx^3 + Dx^4 + \ldots = Ax + \) \[\frac{2B}{1 \cdot 2} x^2 + \frac{2 \cdot 3C}{1 \cdot 2 \cdot 3} x^3 + \frac{24ABC - 8B^3}{1 \cdot 2 \cdot 3 \cdot 4 \cdot A^2} x^4 + \frac{36C^2A^2 + 24ACB^2 - 16B^4}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot A^3} x^5 + \] \[\frac{9 \cdot 24A^2BC^2 - 4 \cdot 24AB^2C^2}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot A^4} x^6 + \frac{216C^3A^3 + 432A^2B^2C^2 - 384ACB^4 + 64B^6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot A^5} x^7 + \ldots \] + &c.; and the series \(1 + qx + rx^2 + sx^3 + tx^4 + \&c = 1 + \frac{B}{A} x + \) \[ \frac{6CA - 2B^2}{1 \cdot 2 \cdot A^2} x^2 + \frac{18CAB - 8B^3}{1 \cdot 2 \cdot 3A^3} x^3 + \frac{36C^2A^2 - 8B^4}{1 \cdot 2 \cdot 3 \cdot 4A^4} x^4 + \] \[ \frac{180C^2A^2B - 120ACB^3 + 16B^5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5A^5} x^5 + \frac{216C^3A^3 + 216A^2C^2B^2 - 288ACB^4 + 64B^6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6A^6} x^6 \] + &c. The terms of these two serieses can easily be deduced by the subsequent method. Let \(Kx^{n-2} + L^{n-1} + Mx^n\), be successive terms of the series \(Ax + Bx^2 + Cx^3 + \&c.\), and \(K_1x^{n-2} + L_1x^{n-1}\) successive terms of the series \(1 + qx + rx^2 + sx^3 + tx^4 + \&c.\); then will \[ M = \frac{2A^2 \times B \times K_1 + 6CAK - 2B^2K}{n \cdot n - 1 \times A^2} \] and \[ L_1 = \frac{n \times A \times M - B \times xL}{A^2}. \] Cor. 1. Let \(B = 0\), and the two serieses \(Ax + Bx^2 + Cx^3 + Dx^4 + \&c.\) and \(1 + qx + rx^2 + \&c.\) become respectively \(Ax + \frac{2 \cdot 3}{2 \cdot 3} Cx^3 + \) \[ \frac{2^2 \cdot 3^2}{2 \cdot 3 \cdot 4 \cdot 5} \times \frac{C^2}{A} x^5 + \frac{2^3 \cdot 3^3}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} \times \frac{C^3}{A^2} x^7 + \frac{2^4 \cdot 3^4}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9} \times \frac{C^4}{A^3} x^9 + \&c., \text{ and } 1 + \frac{2 \cdot 3}{1 \cdot 2} \times \frac{C}{A} x^2 + \frac{2^2 \cdot 3^2}{1 \cdot 2 \cdot 3 \cdot 4} \times \frac{C^2}{A^2} x^4 + \] \[ \frac{2^3 \cdot 3^3}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} \times \frac{C^3}{A^3} x^6 + \&c. \] If in these serieses for \(A\) be substituted \(1\), and for \(C\) be substituted \(- \frac{1}{2 \cdot 3}\), there will result the serieses \(x - \frac{x^3}{2 \cdot 3} + \) \[ \frac{x^5}{2 \cdot 3 \cdot 4 \cdot 5} - \&c., \text{ and } 1 - \frac{x^2}{1 \cdot 2} + \frac{x^4}{1 \cdot 2 \cdot 3 \cdot 4} - \&c. \text{ which give the sine and cosine in terms of the arc } x. \] Cor. 2. Let \(C = 0\), and the above-mentioned series \(Ax + Bx^2 + \&c.\) becomes \(Ax + \frac{2}{1 \cdot 2} Bx^2 * - \frac{2^3}{1 \cdot 2 \cdot 3 \cdot 4} \times \frac{B^3}{A^2} x^4 - \frac{2^4}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} \times \) \[ \frac{B^4}{A^3} x^5 * + \frac{2^6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} \times \frac{B^6}{A^5} x^7 + \frac{2^7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8} \times \frac{B^7}{A^6} x^8 * - \] \[ \frac{2^9}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10} \times \frac{B^9}{A^8} x^{10} - \frac{2^{10}}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11} \times \frac{B^{10}}{A^9} x^{11} + \&c. \text{ The law of } \] this series is, first, that every third term vanishes; and, secondly, the signs of every two successive terms change alternately from + to − and − to +; and, lastly, the co-efficient of the term $x^n$ is $\frac{2^{n-1}}{1 \cdot 2 \cdot 3 \ldots n} \times \frac{B^{n-1}}{A^{n-2}}$; and the series $1 + qx + rx^2 + \&c.$ becomes $1 + \frac{B}{A} x - \frac{2B^2}{1 \cdot 2A^2} x^2 - \frac{2^3B^3}{1 \cdot 2 \cdot 3A^3} x^3 - \frac{2^3B^4}{1 \cdot 2 \cdot 3 \cdot 4A^4} x^4 + \frac{2^4B^5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5A^5} x^5 + \frac{2^6B^6}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6A^6} x^6 + \frac{2^6 \times B^7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7A^7} x^7 + \&c.$ In this series the signs of three successive terms alternately change from + to − and − to +; and the co-efficient of the term $x^n$ is $\frac{2^n \times B^n}{1 \cdot 2 \cdot 3 \cdot nA^n}$ or $\frac{2^{n-1} \times B^n}{1 \cdot 2 \cdot 3 \cdot nA^n}$ according as $n$ is divisible by 3 or not. 2. Let a series $1 + Px + Qx^2 + Rx^3 + Sx^4 + Tx^5 + \&c.$ be of such a formula, that if in it for $x$ be substituted $a + b$, there results a series $1 + P \times a + b + Q \times a + b^2 + R \times a + b^3 + S \times a + b^4 + \&c. = (1 + Pa + Qa^2 + Ra^3 + Sa^4 + \&c.) \times (1 + Pb + Qb^2 + Rb^3 + Sb^4 + \&c.) + (Aa + Ba^2 + Ca^3 + Da^4 + \&c.) \times (Ab + Bb^2 + Cb^3 + Db^4 + \&c.),$ then will the series $Ax + Bx^2 + Cx^3 + Dx^4 + \&c. = Ax + Bx^2 + \left(\frac{2B^2}{3A} - \frac{PB}{3} + A \times \frac{A^2 + P^2}{6}\right) x^3 + \frac{2B^3 - 2PAB^2 + A^2 \times A^2 + P^2 \times B}{6A^2} x^4 + \&c.$, and the series $1 + Px + Qx^2 + Rx^3 + \&c. = 1 + Px + \frac{A^2 + P^2}{2} x^2 + \frac{2AB + P \times A^2 + P^2}{6} x^3 + \frac{4B^2 + A^2 + P^2}{24} x^4 + \&c.$ Let $Kx^{n-2} + Lx^{n-1} + Mx^n$ be successive terms of the series $Ax + Bx^2 + Cx^3 + \&c.$, and $K'x^{n-2} + L'x^{n-1} + M'x^n$ successive terms of the series $1 + Px + Qx^2 + Rx^3 + \&c.;$ then will $A \times L + P \times L' = n \times M'$ and $B \times K + Q \times K' = n \cdot \frac{n-1}{2} \times M'$ express the law of the serieses. Vol. LXXIX. Cor. Let \( B = 0 \), then the series \( Ax + Ex^2 + Cx^3 + Dx^4 = A x \left( x - \frac{P^2 \times A^2}{2 \cdot 3} x^3 + \frac{(P^2 + A^2)^2}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} x^5 + \frac{(P^2 + A^2)^3}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} x^7 + \ldots \right) \), and the series \( 1 + Px + Qx^2 + Rx^3 + \ldots = 1 + Px + \frac{P^2 + A^2}{1 \cdot 2} x^2 + P \times \frac{P^2 + A^2}{1 \cdot 2 \cdot 3} x^3 + \frac{(P^2 + A^2)^2}{1 \cdot 2 \cdot 3 \cdot 4} x^4 + P \times \frac{(P^2 + A^2)^2}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} x^5 + \frac{(P^2 + A^2)^3}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6} x^6 + \ldots \); the coefficient of the term \( x^n \) will be \( (P^2 + A^2)^{\frac{n}{2}} \) or \( P \times (P^2 + A^2)^{\frac{n-1}{2}} \), according as \( n \) is even or odd. If in the equations before given for \( x \) be substituted \( a = b \) instead of \( a + b \), then in the other quantities for \( b \) substitute \( -b \). 3. If in Case 2. the difference between the two quantities \( (1 + Pa + Qa^2 + \ldots) \times (1 + Pb + Qb^2 + \ldots) \) and \( (Aa + Ba^2 + Ca^3 + \ldots) \times (Ab + Bb^2 + Cb^3 + \ldots) \) is assumed \( = 1 + P \times a + b + Q \times a + b^2 + \ldots \), then in the serieses before given for \( A, B, C, \ldots \) write respectively \( \sqrt{-1}A, \sqrt{-1}B, \sqrt{-1}C, \ldots \), and there will result the corresponding serieses. The same principles may be applied to many other cases. 4. Equations of these formulae may be useful, when the sums of the serieses correspondent to a value \( (a) \) of \( x \) are given, and the sums of the series correspondent to a value \( (a + b) \) of \( x \) is required, \( b \) having a small ratio to \( a \): for instance, let the given series be \( x - \frac{x^3}{2 \cdot 3} + \frac{x^5}{2 \cdot 3 \cdot 4 \cdot 5} - \frac{x^7}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \ldots \); the equation found in the first case is \( a + b - \frac{(a + b)^3}{2 \cdot 3} + \frac{(a + b)^5}{2 \cdot 3 \cdot 4 \cdot 5} - \ldots \) \( = (a - \frac{a^3}{2 \cdot 3} + \frac{a^5}{2 \cdot 3 \cdot 4 \cdot 5} - \ldots) \times (1 - \frac{b^3}{1 \cdot 2} + \frac{b^4}{1 \cdot 2 \cdot 3 \cdot 4} - \ldots) \) \(+ (1 - \frac{a^2}{1 \cdot 2} + \frac{a^4}{1 \cdot 2 \cdot 3 \cdot 4} - \ldots) \times (b - \frac{b^3}{2 \cdot 3} + \frac{b^5}{2 \cdot 3 \cdot 4 \cdot 5} - \ldots) \); but but \(a - \frac{a^3}{2 \cdot 3} + \frac{a^5}{2 \cdot 3 \cdot 4 \cdot 5} - \&c.\), and \(1 - \frac{a^2}{1 \cdot 2} + \frac{a^4}{2 \cdot 3 \cdot 4} - \&c.\) are the sine (\(s\)) and cosine (\(c\)) of an arc \(a\) of a circle whose radius is 1; and, consequently, if the sine \(s\) and cosine \(c\) of an arc \(a\) be given, the sine of an arc \((a + b) = s \times (1 - \frac{b^2}{2} + \frac{b^4}{24} - \&c.) + c(b - \frac{b^3}{2 \cdot 3} + \frac{b^5}{2 \cdot 3 \cdot 4 \cdot 5} - \&c.),\) which series, if \(b\) be very small in proportion to \(a\), converges much faster than the common series for finding the sine from the arc: it has been given from different principles in the Meditationes, and is also easily deducible from the series for finding the sine and cosine from the arc by the propositions usually given in plane trigonometry: the cosine of the same arc \((a + b) = c \times (1 - \frac{b^2}{1 \cdot 2} + \frac{b^4}{2 \cdot 3 \cdot 4} - \&c.) - s \times (b - \frac{b^3}{1 \cdot 2 \cdot 3} + \frac{b^5}{1 \cdot 2 \cdot 3 \cdot 5} - \&c.),\) Ex. 2. Let the series be \(a + b + \frac{a + b^3}{2 \cdot 3} + \frac{a + b^5}{2 \cdot 3 \cdot 4 \cdot 5} + \&c.\) \(= (a + \frac{a^3}{2 \cdot 3} + \frac{a^5}{2 \cdot 3 \cdot 4 \cdot 5} + \&c. \times (1 + \frac{b^2}{1 \cdot 2} + \frac{b^4}{1 \cdot 2 \cdot 3 \cdot 4} + \&c.) + (1 + \frac{a^2}{1 \cdot 2} + \frac{a^4}{1 \cdot 2 \cdot 3 \cdot 4} + \&c.) \times (b + \frac{b^3}{1 \cdot 2 \cdot 3} + \frac{b^5}{2 \cdot 3 \cdot 4 \cdot 5} + \&c.);\) but \(a + \frac{a^3}{1 \cdot 2 \cdot 3} + \&c.\) \(= x,\) and \(1 + \frac{a^2}{1 \cdot 2} + \frac{a^4}{1 \cdot 2 \cdot 3 \cdot 4} + \&c.\) \(= \sqrt{1 + x^2},\) if \(a\) be the hyperbolic log. of \(x + \sqrt{1 + x^2};\) therefore \(a + b + \frac{a + b^3}{2 \cdot 3} + \frac{a + b^5}{2 \cdot 3 \cdot 4 \cdot 5} + \&c.\) \(= x \times (1 + \frac{b^2}{2} + \frac{b^4}{2 \cdot 3 \cdot 4} + \&c.) + \sqrt{1 + x^2} \times (b + \frac{b^3}{2 \cdot 3} + \&c.\) Let \(b + \frac{b^3}{2 \cdot 3} + \frac{b^4}{2 \cdot 3 \cdot 4 \cdot 5} + \&c.\) \(= y,\) and \((x + \sqrt{1 + x^2} \times (y + \sqrt{1 + y^2}) = V,\) then will \(a + b + \frac{a + b^3}{2 \cdot 3} + \frac{a + b^5}{2 \cdot 3 \cdot 4 \cdot 5} + \&c.\) \(= \frac{1}{2} V - \frac{1}{V}.\) Ff 2 5. Let a quantity $P$ be a function of $x$, or the fluent of a function of $x \times \dot{x}$, and the value $X$ of it when $x = a$ be known, and the value of it when $x = a + b$ be required. Find a series of which the first term is $X$, and which proceeds according to the dimensions of $b$, if $b$ be a very small quantity, and in general at least so small that the series from $x = a$ to $x = a + b$ neither becomes infinite or 0. In the same manner, if an algebraical or fluxional equation or equations, expressing the relations between $x, y, z, v, \&c.$ be given, find the correspondent values of $y, z, v, \&c.$ to $x = a$, which let be $Y, Z, V, \&c.$; then find serieses for $y, z, v, \&c.$ of which the first terms let be $Y, Z, V, \&c.$ respectively, and which proceed according to the dimensions of $b$, but subject to the same conditions as in the preceding case. From fluxional equations may be deduced series which express the value of $y, \&c.$ in terms of $x$, and always diverge, or always converge, whatever may be its value, as appears from the Meditationes.