420 Mr. Horner 071 [June, 



to find, or approximate toward, the sum of any series whose 

 successive terras are the continual product of quantities in arith- 

 metical prooression. The general value of a,, being — — ; ~ ,/ , 

 we have 



. m+ u r m n (n r — m s) i/ {n + s) „ 



^" 71 + u s n n (?i + M s) n + u s 



— a — p. — "» a _ " (" + '^ ■'■') 



7u- P.. + . f^' - „ + („ +\) s- ^' - n + (» + l)s 'J'' 



in + {u + \) r „ m + r „ u .u + \ .n r—m s-\ r s 



» ~ « + («+!)* ^' 'y" + ' ~ jTTI ^"+ ' '''' ~ n+ (« + l)s.n + (u+2)s 



- M.n + l.fi + 2s.n + 3s ^^ 



^ '^^ '''' ~ 1 .2.n+(u + l)s.n + (u + 2)s '^ 



and so on. The final reduction at each step is preparatory to 

 the next step. Continuing the process till £„, f„, >)„, are found, 

 the law of continuation becomes certain, and then putting y = 1, 

 ■we have 



nr—ms \ n.n+s _ I m + r.n „ 



P' = n.n + s C^' - n + s.n + 2s ^' V' "' "^l ~ n + s.n + 2s ' ^' 



- 2 (n r— ni s + i-s) - n + s.n + 2s », m + 2r,n + s . 



^' = „+2..n + ;is ^'"^'1 ' ■" n+3s.„ + 4s "^i ''' •'"^s ^> " „ + 3*.n + 4s '^^ *' 

 3 (n r— m« + 2r 5)5, n+2s.n + 3s TO + 3r.n + 2s 



&C. &C. 



When these quantities are introduced in their places in Eq. 3, 

 and successive reductions applied on the principle already inti- 

 mated, viz. that any tioo siiccessive numerators, and the denominator 

 belonging to thejirst of them, may be mnltiplied or divided by any 

 common factor yivithout alter ing the value of the series, the value of 



m , m,m + r »n.OT + r.m + 2r , o. i^\ 



« = a . a + a — . a + cx.c. . . (o; 



' n n.n + s n . n + s . n + % s 



becomes in its most disencumbered form. 



a 



r J + — nr—ms , (6) 



n + s + 2(nr — His + rs) , „ , 



M + 2s + — (m + 2r)(n + s) „ , „ . 



n T 4s + . . 



n + as + & 



the law of continuation being manifest. 



This formula (5) includes a large class of useful series, of 

 which the following are a specimen : 



Example I. — Tan."' t, or arc. tan. t = 



t--^f^-^\t^-^t-' + Kc.=^ 



t . t + - ^ , . .t -h Sec. becomes 



3 3.5.7 



