SECT. VI.] DETERMINATION OF THE COEFFICIENTS. 171 



If now we eliminate the last unknown e & by means of the 

 five equations which contain A & , B &) C 5 , D 5 , E.., &c., we find 



a. (5 2 - I 2 ) + 2\ (5 2 - 2 2 ) + 3\ (5 2 - 3 2 ) 

 a 5 (5 2 - I 2 ) + 2 5 5 (5 2 - 2 2 ) + 3 5 c 5 (5 2 - 3 2 ) 

 o 5 (5 2 - I 2 ) + 2 7 5 (5 2 - 2 2 ) + 3V 5 (5 2 - 3 2 ) 



We could have deduced these four equations from the four 

 which form the preceding system, by substituting in the latter 

 instead of 



c 4 , (5 2 -3 2 )c 5 , 



rf 4f (5 2 -4 2 )c/ 5 ; 



and instead of A t , D z A^ B b , 



B t , 5 Jf-C-., 



C 4I 5 (7. -/&amp;gt;., 



By similar substitutions we could always pass from the case 

 which corresponds to a number m of unknowns to that which 

 corresponds to the number m-f-1. Writing in order all the 

 relations between the quantities which correspond to one of the 

 cases and those which correspond to the following case, we shall 

 have 



= c s (5 2 - 3 2 ), rf 4 = rf 5 (5 2 -4 2 ), 



&c ............................ (c). 



