Solutio Generalis Altera Praecedentis Problematis, ope Combinationum & Serierum Infinitarum, per D. Abr. De Moivre. Reg. Soc Sodalem

| Initiat. | Exit. | |----------|-------| | Depositum | krs | | n + 1 | Z | | n + 1 + p | Y | | n + 1 + 2p | X | | n + 1 + 3p | V | | n + 1 + 4p | T | | Depositum | krs | |----------|-------| | n + 1 + p | H | | n + 1 + 2p | K | | n + 1 + 3p | L | | n + 1 + 4p | M | | Depositum | krs | |----------|-------| | n + 1 + p | A = Z | | n + 1 + 2p | C = X | | n + 1 + 3p | D = \(\frac{1}{2}X + \frac{1}{2}Y + yp\) | | n + 1 + 4p | E = \(\frac{1}{2}V + \frac{1}{2}X + yp + \frac{1}{2}Y + zyp\) | \[ \begin{align*} N^o 2 & \\ = \frac{1}{2}H - p + \frac{1}{2}H - p + hp + \frac{1}{8}H - p + 2hp + \frac{1}{16}H - p + 3hp + \cdots \\ & = \frac{1}{2}K - p + \frac{1}{2}H - p + 2hp + \frac{1}{8}H - p + 3hp + \cdots \\ & = \frac{1}{2}L - p + \frac{1}{2}H - p + 3hp + \frac{1}{8}H - p + 4hp + \cdots \\ & = \frac{1}{2}M - p + \frac{1}{2}H - p + 4hp + \frac{1}{8}H - p + 5hp + \cdots \\ & = \frac{1}{2}T - p + \frac{1}{2}H - p + 4hp + \frac{1}{8}H - p + 5hp + \cdots \\ \end{align*} \] \[ \begin{align*} N^o 3 & \\ = \frac{1}{2}C + ncP - cp + \frac{1}{2}D + ndP - dp + \frac{1}{2}E + ncP - cp + \cdots \\ & = \frac{1}{2}D + ndP - dp + \frac{1}{2}E + ncP - cp + \cdots \\ & = \frac{1}{2}E + ncP - cp + \frac{1}{2}F + nfP - fp + \cdots \\ & = \frac{1}{2}F + nfP - fp + \cdots \\ \end{align*} \] \[ \begin{align*} N^o 4 & \\ Z = \frac{1}{2}K - \frac{1}{2}H + \frac{1}{2}hp + \frac{1}{8}hp + \cdots \\ Y = \frac{1}{2}L - \frac{1}{2}K + \frac{1}{2}hp + \frac{1}{8}hp + \cdots \\ X = \frac{1}{2}M - \frac{1}{2}L + \frac{1}{2}hp + \frac{1}{8}hp + \cdots \\ \end{align*} \] \[ \begin{align*} N^o 5 & \\ - H = - \frac{1}{2}C + ncP - cp + \frac{1}{2}D + ndP - dp + \cdots \\ - K = - \frac{1}{2}D + ndP - dp + \cdots \\ - L = - \frac{1}{2}E + ncP - cp + \cdots \\ \end{align*} \] \[ \begin{align*} N^o 6 & \\ C - A = Y - Z = - \frac{1}{2}C - \frac{1}{2}D + \frac{1}{2}E + \cdots \\ X - Y = - \frac{1}{2}D - \frac{1}{2}E + \cdots \\ V - X = - \frac{1}{2}E + \cdots \\ \end{align*} \] \[ \begin{align*} N^o 7 & \\ A \times 2^n + dp \times 2^n - ncP \\ C \times 2^n + cp \times 2^n - ndP \\ D \times 2^n + dp \times 2^n - ncP \\ \end{align*} \] I. Senex sculp. III. Solutio generalis altera praecedentis Problematis, ope Combinati- onum & Serierum infinitarum, per D. Abr. de Moivre. Reg. Soc Sodalem. Designationes. SI B & C collusores duo simul certent, ad designandum B victorem esse, C victum, scribatur BC; atque vi- cissim ad designandum C victorem esse, B victum; scriba- tur CB: & sic de caeteris. Ponatur 1° B vincere A, certamenque concludi tribus ludis BA BC BD Sic patet B victorem necessario evadere. Ponatur 2° B vincere A, certamenque concludi quatuor ludis BA CB CD CA Sic patet C victorem necessario evadere. Ponatur 3° B vincere A, certamenque concludi quinque ludis BA BA CB BC DC DB DA DA DB DC Sic patet D victorem necessario evadere, id- que duplici modo. Ponatur 4° B prima vice vincere A, certamenque concludi sex ludis. Sic patet \( A \) victorem necessario evadere, idque triplici modo. Ponatur 5° certamen concludi septem ludis, ponaturque semper \( B \) prima vice vincere ipsum \( A \). Sic patet \( B \) vel \( C \) necessario victores evadere, \( B \) triplici modo, \( C \) duplici. Ponatur 6° certamen concludi octo ludis, Sic patet \( C \) victorem evadere triplici, \( D \) duplici, \( B \) triplici modo, &c. Nunc ordine scribantur literae quibus victores designantur. \[ \begin{array}{c|c} 3, & 1B \\ 4, & 1C \\ 5, & 2D \\ 6, & 3A \\ 7, & 3B + 2C \\ 8, & 3C + 2D + 3B \\ 9, & 3D + 2A + 3C + 3D + 2A \\ 10, & 3A + 2B + 3D + 2A + 2C + 3D \\ & \text{&c.} \end{array} \] Perspecta illarum formatione, patebit 1° literam B in ordine aliquo semper toties reperiri, quoties A in ordine ultimo & penultimo reperitur: 2° C in ordine aliquo toties reperiri quoties B in ordine ultimo & D in penultimo reperiuntur: 3° D in ordine aliquo toties reperiri quoties C in ultimo & B in penultimo: 4° A in ordine aliquo semper toties reperiri quoties D in ordine ultimo & C in penultimo reperiuntur. Sed numerus variationum dato cuilibet ludorum numero competens, duplus est numeri variationum omnium dato ludorum numero unitate diminuto competentis: adeoque Probabilitas quam habet Collusor B ut vincat dato ludorum numero, est subdupla probabilitatis quam habebat A ut vinceret dato ludorum numero minus uno; atque etiam subquadrupla probabilitatis quam habebat idem A, ut vinceret dato ludorum numero minus duobus: & sic de caeteris. Probabilitas quam habet C, ut vincat dato ludorum numero, est subdupla probabilitatis quam habebat B, ut vinceret dato ludorum numero minus uno; atque etiam subquadrupla probabilitatis quam habebat D, ut vinceret dato ludorum numero minus duobus. Probabilitas quam habet D ut vincat dato ludorum numero, est subdupla probabilitatis quam habebat C, ut vinceret dato ludorum numero minus uno; atque etiam subquadrupla probabilitatis quam habebat B, ut vinceret dato ludorum numero minus duobus. Probabilitas quam habet A ut vincat dato ludorum numero, est subdupla probabilitatis quam habebat D, ut vinceret dato ludorum numero minus uno; atque etiam subquadrupla probabilitatis quam habebat C ut vinceret dato ludorum numero minus duobus. Ex jam observatis facile est componere Tabulam Probabilitatum, quas B, C, D, A habent ut victores evadant dato ludorum numero, atque etiam illorum fortium seu expectationum. Tabula Probabilitatum, &c. | | B | C | D | A | |---|--------------------|--------------------|--------------------|--------------------| | 3 | $\frac{1}{4} \times 4 + 3p$ | | | | | " | | $\frac{1}{8} \times 4 + 4p$ | | | | 5 | | | $\frac{2}{16} \times 4 + 5p$ | | | " | | | | $\frac{3}{32} \times 4 + 6p$ | | v | $\frac{3}{64} \times 4 + 7p$ | $\frac{2}{64} \times 4 + 7p$ | | | | v' | $\frac{3}{128} \times 4 + 8p$ | $\frac{3}{128} \times 4 + 8p$ | $\frac{2}{128} \times 4 + 8p$ | | | v'' | | | | | | v''' | $\frac{4}{512} \times 4 + 9p$ | $\frac{2}{512} \times 4 + 9p$ | $\frac{6}{512} \times 4 + 9p$ | $\frac{4}{512} \times 4 + 9p$ | | x | $\frac{13}{1024} \times 4 + 10p$ | $\frac{10}{1024} \times 4 + 10p$ | $\frac{2}{1024} \times 4 + 10p$ | $\frac{9}{1024} \times 4 + 10p$ | | x' | $\frac{18}{2048} \times 4 + 11p$ | $\frac{19}{2048} \times 4 + 11p$ | $\frac{14}{2048} \times 4 + 12p$ | $\frac{4}{2048} \times 4 + 12p$ | Jam vero Series istae sunt convergentes, adeoque singulæ sum- mari possunt per vulgarem Arithmeticam; & obtinebuntur vel summæ accuratæ si possint, vel saltem approximatæ, si non licet, terminos multos adhibere. Inveni- Invenire summas Probabilitatum ad infinitum usque pergentiam, quas Collusores habent ut victores evadant. Sint Probabilitates omnes ipsius B ad infinitum, nempe \[ B' + B'' + B''' + B'''' + B^v + B^v \&c. = \gamma \] Probabilitates ipsius C \[ C' + C'' + C''' + C'''' + C^v + C^v \&c. = z \] Probabilitates ipsius D \[ D' + D'' + D''' + D'''' + D^v + D^v \&c. = v \] Probabilitates ipsius A \[ A' + A'' + A''' + A'''' + A^v + A^v \&c. = x \] Scribantur autem in Scala perpendiculariter descendente, ad hunc modum. \[ B' = B' \] \[ B'' = B'' \] \[ B' = \frac{1}{2}A'' + \frac{1}{4}A' \] \[ B''' = \frac{1}{2}A''' + \frac{1}{4}A'' \] \[ B^v = \frac{1}{2}A^v + \frac{1}{4}A''' \] \[ B^v = \frac{1}{2}A^v + \frac{1}{4}A'' \] Proinde \( \gamma = \frac{1}{4} + \frac{1}{4}x \). Ergo \( \gamma = \frac{1}{4} + \frac{1}{2}x + \frac{1}{4}x \). Demonstratio. Etenim prima columna perpendicularis \( = \gamma \), ex Hypothesi Est vero \( A' + A'' + A''' + A'''' + A^v \&c. = x \), ex hypothesi; Ergo \( \frac{1}{2}A' + \frac{1}{2}A'' + \frac{1}{2}A''' + \frac{1}{2}A'''' + \frac{1}{2}A^v \&c. = \frac{1}{2}x \). Proinde \( \frac{1}{2}A' + \frac{1}{2}A'' + \frac{1}{2}A''' + \frac{1}{2}A'''' + \frac{1}{2}A^v \&c. = \frac{1}{2}x - \frac{1}{2}A' \). Et \( B' + B'' + \frac{1}{2}A'' + \frac{1}{2}A''' + \frac{1}{2}A'''' \&c. = \frac{1}{2}x - \frac{1}{2}A' + B' + B'' \). Sed Sed $\frac{1}{2} A' = \bar{o}, B'' = o & B' = \frac{1}{4}$, ut patet ex Tabula: Ergo secunda columnna perpendicularis $= \frac{1}{4} + \frac{1}{2} x$. Sed tertia columnna perpendicularis $= \frac{1}{4} x$. erit igitur $y = \frac{1}{4} + \frac{3}{4} x$. Simili modo scribantur $C' = C'$ $C'' = C''$ $C''' = \frac{1}{2} B'' + \frac{1}{4} D'$ $C'''' = \frac{1}{2} B'''' + \frac{1}{4} D''$ hoc est $z = \frac{1}{2} y + \frac{1}{4} v$ $C^v = \frac{1}{2} B^v + \frac{1}{4} D^v$ $C^{v'} = \frac{1}{2} B^{v'} + \frac{1}{4} D^{v'}$ &c. Ergo $z = \frac{1}{8} + \frac{1}{2} y - \frac{1}{8} + \frac{1}{2} v$. Scribantur etiam $D' = D'$ $D'' = D''$ $D''' = \frac{1}{2} C'' + \frac{1}{4} B'$ $D'''' = \frac{1}{2} C'''' + \frac{1}{4} B''$ & pari Argumento patebit $v + \frac{1}{2} z + \frac{1}{4} y$ $D^v = \frac{1}{2} C^v + \frac{1}{4} B^v$ &c. Scribantur denique $A' = A'$ $A'' = A''$ $A''' = \frac{1}{2} D' + \frac{1}{4} C'$ $A'''' = \frac{1}{2} D'''' + \frac{1}{4} C''$ Unde concludetur $x = \frac{1}{2} v + \frac{1}{4} z$ &c. Resolutis autem quatuor istis æquationibus, reperietur \[ B' + B'' + B''' + B'''' \&c. = y = \frac{56}{149} \] \[ C' + C'' + C''' + C'''' \&c. = z = \frac{36}{149} \] \[ D' + D'' + D''' + D'''' \&c. = v = \frac{32}{149} \] \[ A' + A'' + A''' + A'''' \&c. = x = \frac{25}{149} \] Valoribus istis inventis, ponatur jam \( \frac{56}{149} = b, \frac{36}{149} = c, \) \( \frac{32}{149} = d, \frac{25}{149} = a. \) Iterum sit. \[ 3B'p + 4B''p + 5B'''p + 6B''''p \&c. = py. \] \[ 3C'p + 4C''p + 5C'''p + 6C''''p \&c. = pz. \] \[ 3D'p + 4D''p + 5D'''p + 6D''''p \&c. = pv. \] \[ 3A'p + 4A''p + 5A'''p + 6A''''p \&c. = px. \] \[ 3B' = 3E' \] \[ 4B'' = 4B'' \] \[ 5B''' = \frac{1}{2}A'' + \frac{1}{2}A' \] \[ 6B'''' = \frac{1}{2}A''' + \frac{1}{2}A'' \] \[ 7Bv = \frac{1}{2}A'' + \frac{1}{2}A''' \] \[ 8Bv = \frac{1}{2}A' + \frac{1}{2}A'' \] Ergo \( y = \frac{3}{4}x + \frac{1}{4}a \) Etenim prima Columna perpendicularis \( = y \), ex Hypothesi: \[ 3B' + 4B'' = \frac{3}{4}: \text{Nam est } B' = \frac{1}{4}, \& B'' = 0. \] \[ 3A' + 4A'' + 5A''' \&c. = x \text{ ex Hypothesi.} \] \[ A' + A'' + A''' \&c. = a, \text{ut repertum est.} \] Est igitur \( 4A' + 5A'' + 6A''' + 7A'''' \&c. = x + a \) Et \( \frac{1}{2}A' + \frac{1}{2}A'' + \frac{1}{2}A''' + \frac{1}{2}A'''' \&c. = \frac{1}{2}x + \frac{1}{2}a. \) Sed \( A' = 0 \) Ergo secunda Columna perpendicularis \( = \frac{3}{4} + \frac{1}{2} x + \frac{1}{2} a \). \[ \begin{align*} 3A' + 4A'' + 5A''' + 6A'''' &c. = x \\ 2A' + 2A'' + 2A''' + 2A'''' &c. = 2a \end{align*} \] Est igitur \( 5A' + 6A'' + 7A''' + 8A'''' &c. = x + 2a \). Et \( \frac{3}{4}A' + \frac{5}{4}A'' + \frac{7}{4}A''' + \frac{9}{4}A'''' &c. = \frac{1}{4}x + \frac{1}{2}a \). Est igitur tertia Columna perpendicularis \( = \frac{1}{4}x + \frac{1}{2}a \). Erit igitur \( y = \frac{3}{4} + \frac{1}{2}x + \frac{1}{2}a + \frac{1}{4}x + \frac{1}{2}a \) sive \( y = \frac{3}{4} + \frac{1}{2}x + a \), quod erat probandum. \[ \begin{align*} 3C' &= 3C' \\ 4C'' &= 4C'' \\ 5C''' &= \frac{5}{2}B'' + \frac{5}{2}D' \\ 6C'''' &= \frac{6}{2}B''' + \frac{6}{2}D'' \\ 7C''''' &= \frac{7}{2}B'''' + \frac{7}{2}D''' \\ 8C'''''' &= \frac{8}{2}B''''' + \frac{8}{2}D''''' \end{align*} \] &c. Ergo \( z = \frac{1}{2}y + \frac{1}{2}b + \frac{1}{4}v + \frac{1}{2}d \). Etenim prima Columna perpendicularis \( = z \), ex Hypothesi. \[ \begin{align*} 3C' + 4C'' &= \frac{1}{2} \\ 3B' + 4B'' + 5B''' + 6B'''' &c. = y \\ B' + B'' + B''' + B'''' &c. = b \end{align*} \] Est igitur \( 4B' + 5B'' + 6B''' + 7B'''' &c. = y + b \). Sed \( 4B' = 1 \). Ergo \( 5B'' + 6B''' + 7B'''' &c. = y + b - 1 \). \[ \begin{align*} \frac{5}{2}B'' + \frac{5}{2}B''' + \frac{5}{2}B'''' &c. = \frac{1}{2}y + \frac{1}{2}b - \frac{1}{2} \end{align*} \] Ergo secunda Columna perpendicularis \( = \frac{1}{2} + \frac{1}{2}y + \frac{1}{2}b - \frac{1}{2} = \frac{1}{2}y + \frac{1}{2}b \). Iterum, \( 3D' + 4D'' + 5D''' + 6D'''' &c. = v \) \[ \begin{align*} 2D' + 2D'' + 2D''' + 2D'''' &c. = 2d \end{align*} \] Est igitur \( 5D' + 6D'' + 7D''' + 8D'''' &c. = v + 2d \). Et \( \frac{5}{4}D' + \frac{5}{4}D'' + \frac{5}{4}D''' + \frac{5}{4}D'''' &c. = \frac{1}{4}v + \frac{1}{2}d \). Ergo tertia Columna perpendicularis \( = \frac{1}{4}v + \frac{1}{2}d \) Est igitur \( z = \frac{1}{2}y + \frac{1}{2}b + \frac{1}{4}v + \frac{1}{2}d \), quod erat probandum. Eodem prorsus ordine scribantur. \[ \begin{align*} 3D' & = 3D' \\ 4D'' & = 4D'' \\ 5D''' & = \frac{1}{2}C'' + \frac{1}{4}B' \\ 6D'''' & = \frac{1}{2}C''' + \frac{1}{4}B'' \\ 7D'''' & = \frac{1}{2}C'''' + \frac{1}{4}B''' \\ 8D'''' & = \frac{1}{2}C'''' + \frac{1}{4}B'''' \\ & \vdots \\ 3A' & = 3A' \\ 4A'' & = 4A'' \\ 5A''' & = \frac{1}{2}D'' + \frac{1}{4}C' \\ 6A'''' & = \frac{1}{2}D''' + \frac{1}{4}C'' \\ 7A'''' & = \frac{1}{2}D'''' + \frac{1}{4}C''' \\ 8A'''' & = \frac{1}{2}D'''' + \frac{1}{4}C'''' \\ & \vdots \\ \end{align*} \] Unde \(v = \frac{1}{2}x + \frac{1}{2}c + \frac{1}{4}y + \frac{1}{2}b\). Et \(x = \frac{1}{2}v + \frac{1}{2}d + \frac{1}{4}z + \frac{1}{2}c\). Quae quidem Conclusiones eodem modo demonstrantur ac superiores. Solutis autem quatuor istis aequationibus, elicetur \[ \begin{align*} y & = \frac{45536}{149^2}, \\ z & = \frac{38724}{149^2}, \\ v & = \frac{37600}{149^2}, \\ x & = \frac{33547}{149^2} = \frac{33547}{22201}. \end{align*} \] Ergo, si velint \(B, C, D, A\) vendere Spectatori cuidam \(R\) summas quas singuli obtinere sperant, æquum erit ut emptor \(R\) pendat \[ \begin{align*} \text{ipsi } B & = 4 \times \frac{56}{149} + \frac{45536}{22201} p, \\ \text{ipsi } C & = 4 \times \frac{36}{149} + \frac{38724}{22201} p, \\ \text{ipsi } D & = 4 \times \frac{32}{149} + \frac{37600}{22201} p, \\ \text{ipsi } A & = 4 \times \frac{25}{149} + \frac{33547}{22201} p. \end{align*} \] Invenire Probabilitates quas habent \(B, C, D, A\), ut multentur, dato ludorum numero. Si Ludi duo tantum sint, erunt hoc modo. \[ \begin{align*} BA & = BA \\ CB & = BC \end{align*} \] Unde patet \(B\) vel \(C\) necessario multari. Si Ludi tres fuerint, hoc modo se res habet. \[ \begin{align*} BA & = BA \\ CB & = CB \\ DC & = CD \\ BA & = BA \\ BC & = BC \\ BD & = BD \end{align*} \] Hinc patet \(C\), vel \(D\) vel \(B\) necessario multari. B b Si Si vero quatuor Ludi fuerint: \[ \begin{array}{cccccc} BA & BA & BA & BA & BA & BA \\ CB & CB & CB & CB & BC & BC \\ DC & DC & CD & CD & DB & DB \\ AD & DA & AC & CA & AD & DA \end{array} \] Debet igitur \(A\) triplici modo, \(D\) duplici, \(C\) simplici, \(B\) muletari. Et sic de cæteris. Ex quibus manifesta est Compositio Tabulae subjunctæ Probabilitatum quas \(B, C, D, A\) habent ut muletentur, dato ludorum numero. | Num Lud. | B | C | D | A | |----------|-----|-----|-----|-----| | 1 | 1/2 | 1/2 | | | | 2 | 1/4 | 1/4 | 1/4 | | | 3 | 1/8 | 1/8 | 1/8 | 1/8 | | 4 | 1/16| 1/16| 1/16| 1/16| | 5 | 1/32| 1/32| 1/32| 1/32| | 6 | 1/64| 1/64| 1/64| 1/64| Sint autem \(y, z, v, x\) summæ omnium Probabilitatum quas \(B, C, D, A\) habent respective ut muletentur. Scribantur eodem ordine ac in praecedentibus. \[ \begin{align*} B' &= B' \\ B'' &= B'' \\ B''' &= \frac{1}{2}A' + \frac{1}{4}A' \\ B'''' &= \frac{1}{2}A'' + \frac{1}{4}A'' \\ B'''''' &= \frac{1}{2}A''' + \frac{1}{4}A''' \\ B'''''''' &= \frac{1}{2}A'''' + \frac{1}{4}A'''' \\ &\vdots \\ C' &= C' \\ C'' &= C'' \\ C''' &= \frac{1}{2}B' + \frac{1}{4}D' \\ C'''' &= \frac{1}{2}B'' + \frac{1}{4}D'' \\ C'''''' &= \frac{1}{2}B''' + \frac{1}{4}D''' \\ C'''''''' &= \frac{1}{2}B'''' + \frac{1}{4}D'''' \\ &\vdots \\ Ergo y = \frac{3}{4} + \frac{1}{2}x + \frac{1}{4}x. \\ Ergo z = \frac{1}{2} + \frac{1}{2}y + \frac{1}{4}v. \\ \end{align*} \] Scribantur deinde \[ D' = D' \] \[ D'' = D'' \] \[ D''' = \frac{1}{2} C'' + \frac{1}{4} B' \] \[ D'''' = \frac{1}{2} C'''' + \frac{1}{4} B'' \] \[ D'''' = \frac{1}{2} C'''' + \frac{1}{4} B'''' \] \[ D'''' = \frac{1}{2} C'''' + \frac{1}{4} B'''' \] \[ A' = A' \] \[ A'' = A'' \] \[ A''' = \frac{1}{2} D' + \frac{1}{4} C' \] \[ A'''' = \frac{1}{2} D'''' + \frac{1}{4} C'' \] \[ A'''' = \frac{1}{2} D'''' + \frac{1}{4} C'''' \] \[ A'''' = \frac{1}{2} D'''' + \frac{1}{4} C'''' \] \[ \text{etc.} \] Ergo \( v = \frac{1}{4} + \frac{1}{2} z + \frac{1}{4} y. \) Ergo \( x = \frac{1}{2} v + \frac{1}{4} z. \) Resolutis autem quatuor istis æquationibus, invenietur \[ y = \frac{243}{249} \] \[ z = \frac{252}{149} \] \[ v = \frac{224}{149} \] \[ & x = \frac{175}{149} \] Ergo si velit Spectator aliquis \( S \) multas omnes sustinere, æquum erit ut ipsi \( S \) \[ B \text{ tradat } \frac{243}{149} p \] \[ C \text{ tradit } \frac{252}{149} p \] \[ D \text{ tradit } \frac{224}{149} p \] \[ & A \text{ tradit } \frac{175}{149} p. \] Sublatis itaque summis probabilitatum quas singuli Collusores habent ut multiplicentur, è summis expectationum quas habent iudem si victores abeant, restabunt sortes eorum respective : nempe \[ B \text{ recipit ab } R \frac{4 \times 56}{149} + \frac{45536}{22201} p \] \[ B \text{ tradit ipsi } S \frac{243}{149} p \] Ergo ipsi \( B \) superest \( \frac{224}{149} + \frac{9329}{22201} p \) Sed \( B \) depoluerat \( 1 \) priusquam ludus inciperet. Ergo \( B \) lucratur \( \frac{75}{149} + \frac{9329}{22201} p. \) C recipit ab R $\frac{4 \times 36}{149} + \frac{38724}{22201} p$ C tradit ipsi S $\frac{252}{149} p$ Ergo ipsi C superest $\frac{144}{149} + \frac{1176}{22201} p$ Sed C deposuerat i. Ergo C lucratur $-\frac{5}{149} + \frac{1176}{22201} p$. D recipit ab R $\frac{4 \times 32}{149} + \frac{37600}{22201} p$ D tradit ipsi S $\frac{224}{149} p$ Ergo ipsi D superest $\frac{128}{149} + \frac{4224}{22201} p$ Sed D deposuerat i. Ergo D lucratur $-\frac{21}{149} + \frac{4224}{22201} p$. A recipit ab R $\frac{4 \times 25}{149} + \frac{33547}{22201} p$ A tradit ipsi S $\frac{175}{149} p$ Ergo ipsi A superest $\frac{100}{149} + \frac{7472}{22201} p$ Sed A deposuerat i + p, nempe i priusquam ludus inchoaretur, & p postquam semel victus fuerat à B: Ergo A lucratur $-\frac{49}{149} - \frac{14729}{22201} p$. Lucrum ipsius $B = + \frac{75}{149} + \frac{9329}{22201} p$ ipsius $C = - \frac{5}{149} + \frac{1176}{22201} p$ ipsius $D = - \frac{21}{149} + \frac{4224}{22201} p$ ipsius $A = - \frac{49}{149} - \frac{14729}{22201} p$ Summa Lucrorum $= 0$ Summa autem lucrorum ipsorum $B \& A = \frac{26}{149} - \frac{5400}{22201} p$; sed posueramus $B$ vicisse ipsum $A$ semel, priusquam Collusores pacta inirent cum $R \& S$. Priusquam vero ludus inchoaretur, $A$ poterat æqua sorte expectare ut vinceret ipsum $B$, adeoque summa lucrorum $\frac{26}{149} - \frac{5400}{22201}$ in duas partes æquales est dividenda, ita ut utriusque lucrum censendum sit $\frac{13}{149} - \frac{2700}{22201} p$. Ponatur $\frac{13}{149} - \frac{2700}{22201} p = 0$, & erit $p = \frac{1937}{2700}$. Ergo si sit mulæta $p$ ad summam quam singuli deponunt ut $1937$ ad $2700$, $A \& B$ nihil lucrantur, nihil perdunt. Verum hoc in Casu $C$ lucratur $\frac{1}{225}$, quam $D$ perdit. Coroll. 1. Spectator $R$, priusquam ludus inchoetur, id suscipere in se poterit, ut summam $4$ de qua Collusores contendunt, & mulætas omnes pendat, si sibi initio in manus darentur $4 + 7 p$. Coroll. 2. Si dexteritates Collusorum sint in ratione data, sortes Collusorum eadem ratiocinatione determinabuntur. Cc Coroll. Coroll. 3. Si Series aliqua ita sit constituta, ut continuò decrecat, & terminus quivis ad præcedentes quoslibet habeat rationes datas, sive easdem sive diversas, series ista accurate summabitur. Insuper si termini omnes hujus Seriei multiplicentur per terminos progressionis Arithmeticae, singuli per singulos, Series nova resultans accurate summabitur. Coroll 4. Si sint Series plures collaterales, ita relatae ut terminus quilibet cujusque Seriei ad præcedentes quoslibet aliarum Serierum habeat rationes datas, sive easdem sive diversas, ita ut Series istae collaterales se decussent data qualibet lege constanti, Series istae accurately summabuntur. Insuper si termini omnes harum Serierum multiplicentur ordinatim per terminos Progressionis Arithmeticae, singuli per singulos, Series novae ex hac multiplicatione resultantes etiamnum accurate summabuntur. Clavis ad Problema generale. Si sint Collusores quotcunque v.g. Sex, B, C, D, E, F, A & Probabilitates quas habent ut victores evadant, sive ut multentur, dato Ludorum numero, denotentur respective B, C, D, E, F & A; & Probabilitates dato Ludorum numero his proximo & minori competentes, per B,, C,, D,, E,, F,, A,,; & Probabilitates dato Ludorum numero his itidem novissimis proximo & minori competentes, per B,,, C,,, D,,, E,,, F,,, A,,, & sic deinceps; erit semper, \[ B_i = \frac{1}{2}A_{ii} + \frac{1}{4}A_{iii} + \frac{1}{8}A_{iv} + \frac{1}{16}Av \] \[ C_i = \frac{1}{2}B_{ii} + \frac{1}{4}F_{iii} + \frac{1}{8}E_{iv} + \frac{1}{16}Dv \] \[ D_i = \frac{1}{2}C_{ii} + \frac{1}{4}B_{iii} + \frac{1}{8}F_{iv} + \frac{1}{16}Ev \] \[ E_i = \frac{1}{2}D_{ii} + \frac{1}{4}C_{iii} + \frac{1}{8}B_{iv} + \frac{1}{16}Fv \] \[ F_i = \frac{1}{2}E_{ii} + \frac{1}{4}D_{iii} + \frac{1}{8}C_{iv} + \frac{1}{16}Bv \] \[ A_i = \frac{1}{2}F_{ii} + \frac{1}{4}E_{iii} + \frac{1}{8}D_{iv} + \frac{1}{16}Cv \] Et fiat semper retrogressus ordinatim ad tot literas minus duobus quot sunt Collusores, omittaturque semper litera A, prima æquatione excepta, ubi litera A terminos omnes praeter primam occupat.