150 THE LAWS OF 



the equation immediately preceding will become 



x ft (0).I7 ' i >+4>(l).^'V+4>(2) tf 2 >.r<-f &c.) 



sm 



which compared with the foregoing value of V (i \ will give by 

 suppressing the factor *"" ' '_ , common to both and equat- 



ing separately the coefficients of the different powers of the inde- 

 terminate quantity r the following system of equations 



A ~ 4 n *S n , (i) 4-n. 6 n.S n , {i) 



2.4 ** ' 2.4.6 ~ J* 



~ " 



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



for determining the unknown functions / (i) , f, f\ &c. by 

 means of the known ones U Q (i \ U, Z7 2 (i) , &c. In fact the last 

 equation of the system gives U s (i} =f 8 (i) , and then by ascending 

 successively from the bottom to the top equation, we shall get 

 the values of / 8 (i) , /, ( . i) 1 , j/jw , &c. with very little trouble, it will 

 however be simpler still to remark, that the general type of all 

 our equations is 



where the symbols of operation have been separated from those 

 of quantity and e employed in its usual acceptation, so that 



But it is evident we may satisfy the last equation by making 



Expanding now and replacing eZ7 tt (i) , e 2 Z7 tt (i) , &c. by these 

 values R&, RiS,&c.weget 



