138 THEORY OF HEAT. [CHAP. III. 



sarily satisfy the equations which are obtained by successive 



differentiations ; whence the following results, 



1 = a cos y + b cos 3y + c cos 5y + d cos 1y -f &c., 

 = a sin y + 3b sin 3y + 5c sin 5y + 7d sin 7y + &c., 

 = a cos y + 3 2 & cos 3# + 5 2 c cos 5^ + 7 2 c cos 7?/ + &c., 



= a sin y + 3 3 6 sin 3y + 5 3 c sin oy + Td sin 7y + &c., 

 and so on to infinity. 



These equations necessarily hold when y = 0, thus we have 



1 = a+ 5+ c+ cl+ e+ f+ 0+...&C., 

 = a + 3 2 t&amp;gt; + 5 2 c + 7 2 d^ + 9 2 e + H 2 /+ ... &c., 



= a + 3 4 5 + 5 4 c + 7 4 J+9 4 6+ ... &c., 



= a + 3 6 6 + 5 G c + 7 6 ^+ ... &c., 



= a + 3 8 6 + 5 8 c -f . . . fec., 



&c. 



The number of these equations is infinite like that of the 

 unknowns a, b, c, d, e, ... &c. The problem consists in eliminating 

 all the unknowns, except one only. 



172. In order to form a distinct idea of the result of these 

 eliminations, the number of the unknowns a, b, c, d, ...&c., will 

 be supposed at first definite and equal to m. We shall employ 

 the first m equations only, suppressing all the terms containing 

 the unknowns which follow the m first. If in succession m 

 be made equal to 2, 3, 4, 5, and so on, the values of the un 

 knowns will be found on each one of these hypotheses. The 

 quantity a, for example, will receive one value for the case 

 of two unknowns, others for the cases of three, four, or successively 

 a greater number of unknowns. It will be the same with the 

 unknown 6, which will receive as many different values as there 

 have been cases of elimination ; each one of the other unknowns 

 is in like manner susceptible of an infinity of different values. 

 Now the value of one of the unknowns, for the case in which 

 their number is infinite, is the limit towards which the values 

 which it receives by means of the successive eliminations tend. 

 It is required then to examine whether, according as the number 

 of unknowns increases, the value of each one of a, b, c, d ... &c. 

 does not converge to a finite limit which it continually ap 

 proaches. 



