148 THE LAWS OF 



/ O ' (l) > /i' (i) > ^' (i) ' & c - being quantities independent of r and all of 

 the form Y' (i) (Mec. Gel Liv. in.). Moreover V t (i} the part of V 

 due to the general term f^r' Mt of the last series, will be ob- 



tained by writing n ~ for /3 in the equation (11), and after- 

 wards substituting for 



In this way we get 



* * *? 



2.4...2*+2i"-l 

 sm 



n-l.n+1 ...w + 2i'4-2*'-3 

 x 



2. 4. ..2*' 



f t (i] being what j^ becomes by changing 6', TV into 8, -or, and 

 the finite integral being taken from t' to t' = co . 



Let us now for a moment assume. 



4 n I .n+1 ... n + Si+St' S 



y ! 



^ ^Bf ,- k .,-v.f.. 5 



2 . 4 . . . 2t' 



then the expression immediately preceding may be written 

 2-7T 2 . /J (i) .r* ^4 w.6-n... 2^ 2^' + 2 w 



sm 



, t] w 

 T\ I'* 



and by giving to t the various values 0, 1, 2, 3, &c. of which it 

 is susceptible, and taking the sum of all the resulting values of 

 V t (i] the quantity thus obtained will be equal to F (i) or that part 

 of V which is of the form Y (i} . Thus we get 



