172 THEORY OF HEAT. [CHAP. III. 



We have also 



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



From equations (c) we conclude that on representing the un 

 knowns, whose number is infinite, by a, b, c, d, e, &c., we must 

 have 



a 



(3* - 2 2 ) (4 2 - 2 2 ) (5 2 - 2 2 ) (6 2 - 2 2 ) . . . 



~ (4 a - 3 2 ) (5 2 - 3 2 ) (6 2 - 3 2 ) (T - 3 2 ) . . . 



d = (5* _ 4 ) (G 2 - 4 2 ) (T - 4 2 ) (8 2 - 4 2 ) . . . 



&c. &c (e). 



209. It remains then to determine the values of a lt 6 2 , c 8 , 

 d 4 , e e , &c. ; the first is given by one equation, in which A enters; 

 the second is given by two equations into which A 2 B Z enter; the 

 third is given by three equations, into which A 3 B 3 C 3 enter ; and 

 so on. It follows from this that if we knew the values of 



A 19 A 2 B 2 , A 3 B 3 C 3 , Af^CJ),..., &c., 



we could easily find a x by solving one equation, a 2 & 2 by solving 

 two equations, a 3 b 3 c 3 by solving three equations, and so on : after 

 which we could determine a, b } c, d, e, &c. It is required then 

 to calculate the values of 



..., &c, 



by means of equations (d). 1st, we find the value of A 2 in 

 terms of A % and 5 2 ; 2nd, by two substitutions we find this value 

 of A 1 in terms of A 3 B 3 C 3 ; 3rd, by three substitutions we find the 



