ALGEBRAIC FORMULA. 



257 



and fiibftitution of A for ^-^3 — , and of cof (/> + i)<p+ cof (/> — i )<p 

 for 2 cof <p cofp<p, we get the following equation, 

 AA' +AB'1 +AC'-1 +AD"1 



--2 A' I — B' I — C I 



ni Qi }>C0f(p j-j/ }»Cof2(p £/ |>cof 3(p, &c. = o. 



A 



' ab 



ab 



^' 1 



ab J 



Hence it follows, that, 



B' = AA' 

 C = AB' 

 D' = AG' 



E 

 D 



'a&. -> 



A 

 'ab 



■ ^ Tib 



no 



E' = AD' 



&c. &c. 



It yet remains to determine A', the firft coefficient ; but that, 

 as well as B', may be readily found by confidering that we have 

 already found (Art 4.) 



therefore, •fubftitutlng « — i for «,, and C for C, for the very 

 fame reafon we get 



C'= -4-B'- 



n + I 



2>t — 2 



n + I 



A'. 



. By corriparing together this value of C, and the values of B' 

 and C as given in the laft feries of equations, and fubftituting^ 



for A its value ^^ , we get, 



A' 



a' +h' 



■ (a^ — b^y 





•R'___4fi_ A. 



(« + i) («' + ^') 



B. 



6. It 



