174 THE LAWS OF 



yn-l.n+l...n + 2t'-3 n- 1 .n + 1 ... n + 2t'+ 2*'- 3 

 r 2 . 4 ... 2t' 2.4 ... 2i+2t' 



3-n.5-n... l + 2t-2t'-n 



the sign of integration 2 belonging to the variable t'. 



Having thus the part of V due to the term a t (i) cos iff in the 

 expansion of /(a/, y r ) it is clear that we may thence deduce the 

 part due to the analogous term b t (i} sin iff by simply changing 

 a t M cos id into b t M sin i&, and consequently we shall have the 

 total value of V itself, by taking the sum of the various parts 

 due to all the different terms which enter into the complete ex- 

 pansion of /(a?', y'). 



If now we make /3 = - and recollect that 



sm -_ 



the foregoing expression will undergo simplifications analogous 

 to those before noticed (Art. 5). Thus we shall obtain 



sm (-3- 



3 3-^. 5 -n...l +2t- 2t'-n 



2.4... 2^+2*' 2.4 ... 



or by writing for abridgment 

 ... ,^_ 



2 . 4 2.4 ... 



there will result this particular value of /3 



and afterwards by making 



