352 Outline of general Methods for the Development 



This may easily be accomplished by many means when the 

 charactei- of (p^^ z is slightly known. In pointing out one means 



I take advantage of the labours of Spence. ^^^e can state the 

 expanded expression 



(p z + (p ~~' = 2^ .l+2<p .l.i^Sf' 



2m + l 2»i + l 2OT-J-1 Im-X ^ • '^ 



' r ■ * \ o 1 A ^ 

 2m — 2 



1.2.3.4 



log: 



+2^.-1 



[2m] 



And if z be made = — 1, and a substitution made for log — 1 

 in terms of John Benouilli's quadrature of the circle, which 

 might be deduced fiom the imaginary expressions for the sine 

 and cosine, we shall have a particular series in terms off .1, 

 f —\ and TT. If again ip z be supposed to consist either of 



odd powers or of even powers of z (which it may be seen 

 from the constitution of it will do, if (p^ z consists of even or 

 odd powers), then will be given a very simple relation betwixt 

 <p 1 and p — 1. Let it consist of odd powers, and we shall 



then have ^o 1 = c \. ^ - p\ 



27n + l 2OT-1 ' -^ 2»i-3 ^•2-3-4 



+ <P I'-To-o-^^ - ± ?.l 



2,,^_. 1.2.3.4.5.6 -- ^2^^^2n^i 



whence we may calculate 



I "^ 



5J-1 



1.2.3.4 



17ip,l 



^'- 1.2.3.4.5.6 4 



^•1 = 1.2. 3.4^5.6 — 8-^^'^-^- 

 &c. &c. 



I shall afterwards have occasion to give a simple and com- 

 modious formula for the general determination of <p^ 1 in such 

 series, in terms of (p, 1 ; so that I am saved the necessity of 

 digressing at this period of the paper. It is sufficient in the 

 mean time to see that the constants may be determined, and 

 of course that the quantity 



M (c + 4') is given. 



Let us now seek the illustration of a particular case, f z 



will 



