426 Mr green, ON THE DETERMINATION OF THE 



To satisfy this equation, let us assume 



Then by substituting in the above and equating separately the coefficients 

 of the various powers of yu, we have in the first place from the highest 



X._, = - e{e^-r—\) (44), 



and afterwards generally 



. e-i-9.t .e-i-M-\ . 



'*' ~ ~ 2/ + 2x2e + r-2#-3 " 



But the equation (43) may evidently be made to coincide with (44), by 

 writing «*''' for i, and t^''+'^ for e, since then both will be comprised in 



\,_,+, = - e*--' {e<'-* + r-2| (45). 



Hence we readily get for the general solution of the system (41), 



"^ 2.4 X {2f<'-> + r-3|{2«"-' + r-5} " - &C.J ; 



where w = cos 9,_r, and i*''* represents any positive integer whatever, pro- 

 vided ^''■' is never greater than ^*'■*". 



Though we have thus the solution of every equation in the system 

 (41), yet that of the first may be obtained under a simpler form by 

 writing therein for X^.i its value — i® deduced from (45). We shall 

 then immediately perceive that it is satisfied by 



cos [ J 



In consequence of the formula (45), the equation (42) becomes 



^- dp' ^ pO-p') dp \ / '^T^'i^' 



which is satisfied by making ^= —\, -(«'*' + 2ft)) (e"*'' + 2a) + w — l), and 



