174 THEORY OF HEAT. [CHAP. ITT. 



now the quantities P lt Q lf E lt S lt T^... &c. may be easily deter 

 mined, as we shall see lower down ; hence the first coefficient a 

 becomes entirely known. 



210. We must pass on now to the investigation of the follow 

 ing coefficients b, c, d, e, &c., which from equations (e) depend on 

 the quantities 6 2 , c 3 , d 4 , e s , &c. For this purpose we take up 

 equations (6), the first has already been employed to find the 

 value of ffj, the two following give the value of 6 2 , the three 

 following the value of C 3 , the four following the value of d 4 , and 

 so on. 



On completing the calculation, we find by simple inspection 

 of the equations the following results for the values of 6 2 , c s , r7 4 , 

 &c. 



3c 3 (I 2 - 3 2 ) (2 2 - 3 2 ) = A 3 l 2 . 2 2 - B z (I 2 + 2 2 ) + &amp;lt;7 3 , 

 4&amp;lt;Z 4 (l 2 -4 2 )(2 2 -4 2 )(3 2 -4 2 ) 



= .4 4 l 2 . 2 2 . 3 2 -^ 4 (I 2 . 2 2 + I 2 . 3 2 + 2 2 .3 2 ) + C 4 (1 2 + 2 2 -f 3 2 ) -7&amp;gt; 4 , 

 &c. 



It is easy to perceive the law which these equations follow ; 

 it remains only to determine the quantities A n B n , A 2 B 3 C 3 , 

 A$f!v &c. 



Now the quantities A. 2 B 2 can be expressed in terms of A 3 B 3 C 3 , 

 the latter in terms of A 4 B 4 C 4 D 4 . For this purpose it suffices to 

 effect the substitutions indicated by equations (d) ; the successive 

 changes reduce the second members of the preceding equations 

 so as to contain only the AB CD, &c. with an infinite suffix, 

 that is to say, the known quantities ABCD, &c. which enter into 

 equations (a) ; the coefficients become the different products 

 which can be made by combining the squares of the numbers 

 1*2*3*4*5* to infinity. It need only be remarked that the first 

 of these squares I 2 will not enter into the coefficients of the 

 value of a t ; that the second 2 2 will not enter into the coefficients 

 of the value of b. 2 ; that the third square 3 2 will be omitted only 

 from those which serve to form the coefficients of the value of c 3 ; 

 and so of the rest to infinity. We have then for the values of 



