Mr. Jarrett on Algehraic Notation. 



79 



= o—nS, 



.-. (-1) 



r . s Ar - s . n . — 



11 V \ ^ 



{r-s) 



(Art. 11.) 



nr.Sr^-, — h— .\r-\ .\r~s.n. 



' S- 1 ' ; l-r 



m 



(r-s) 



:.f[r, t) = \m + (nr-t) ./{r, t-\) -\m_. [nr-t) 

 But \m - \m.{nr — t) — n (— - r) 



711 



f+i t 



= [m {m - 1 — nr ^ t — m-\-nr\ = 0; 



t 



.■.f{r,t)= (nr-t).f{r,t-l)+ (m-nr) .f{r- 1, t- \) . . . .(2). 

 For t, put t - I, then 



f{r,t- 1) = {nr-t+l)f{r,t-2) + {m-nr)f(r-l,t-2). 

 For r and t put r — I and ^ — 1 respectively, then 



f{r~l,t-l) = (.n.r-l-t + l)/(r-l,t-2)+{m-n.r-l)f{r-2,t-2) 



Put ^,. , = coefficient of /(7--r, < — s) in /{>; t), and we get 

 /{r, t) ^ J,,, ./ (r, t- I) + (m-m-) ./ {r - 1, t-l) 

 = A„,,./{r,t-2) + J,,,.f{r-l,t-2) 



+ {ni — nr] {m — n.r-\).f{r — 2,t-2) 

 = ^„,,./(r, <-2) +A,_,.f{r-\,t-2) 



m 



+ n' 



n 



-r.f{r-2,t-2) 



= S\.A,.,,..f{r+ l-v,t-s) 



+ n' 



m 

 n 



-r.f{r-s,t-s). 



