158 THE LAWS OF 



Giving now to t the successive values 0, 1, 2, 3, &c. and 

 taking the sum of the functions thence resulting, there arises 



V= F (0) = F (0) + V + F 2 (0) + F 3 (0) + etc. = 8. V 



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



^ 



2 



where the sign 8 is referred to the variable t arid 2 to t'. 



Again, by substituting for F and V their values in the 

 equation V + V const, and expanding the function V we 

 obtain 



'n2 

 sm( 



y 4, 

 * r 



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



which by equating separately the coefficients of the various, 

 powers of the indeterminate quantity r, and reducing, gives 

 generally 



2 w . 4 n ...2s n 

 ~2 77. 2s 



Then by assigning to ' its successive values 1, 2, 3, &c., there 

 results for the determination of the quantities B Q , B^ B^ &c., 

 the following system of equations, 



&c ........ &c. . . . &c. . 



