1826.] Solutions of the Function i>' x. 171 



Whence 



_ sin.jrS- / -fx.-i sin. 8&. A"-' 



*' - sin.a *^ ^ -sin.S(2cos.Sr- (iy) 



and this again produces 



__ sm. (z + l)& . VB _ sin. (j + 1)5 . A 

 "'■ sin. z 3- ~ sin. 2 & . 2 cos.* 



The latter expression, in a more convenient form, becomes 



_ sin, (t + l) a . (J + c) 

 "'■ ~ sin. (j + 1) * + sin. (z-1) & (20) 



In the mean time, the original assumption in equation (18), 

 when resolved produces 



7 _ —(.b* — 2 COS. 2 & . J c + c") 



~ (2 + 2 COS. 2 b) a (2^) 



But there is another quantity, which we may denote by » 

 and which is connected to the rest by the equations 



P. + q. = r + p, and^, 5'_, = rp (22) 



which in the case we are now examining becomes 



_ sin.(i- !)&.(» + c) 



^■" ~ sin. (z + 1) & + sin, (?-!)& C^^) 



If these values are substituted in the formulge 



•^ (S= - 4) + d X {c - p,) + d.v V-^^^ 



which are readily seen to be convertible expressions, though one 

 of them may occasionally be more convenient than the other, 

 they ^complete the solution in the case of imaginary roots, or 4 B 



6. From these general formulae, the periodic species are readily 

 derived and in a more natural and perspicuous manner than 

 from the mdependent trains of reasoning which have usually 

 been employed. 



The general form, then, of the periodic species concerned in 

 this inquiry, being ^I,"a; = x, which flows from/" x = x it is 

 obvious from Equation 8, that the general condition of solution 



IS 



s„ = 0,1. e. ^ = 



r - p 



Here, if r - p = 0, .9.. is = « ,-"-• .-. s, = 0, which leads to 

 the truismyo x = x, and is no solution. 



If ;• + p = 0, which is a divisor of s„ only when « is an even 

 number, we have r + p = A = b + c = ; and, therefore, 



fx = 



a — c X 



c + d X 



which, as we have already seen (Art. 3), is the ;jro;>er solution of 



J X — X. It is but incidentally a solution in the general event 



of » = 2 w, on a well-known principle common to°all the cases 



