176 THEORY OF HEAT. [CHAP. III. 



212. If we consider now equations (e) which give the values 

 of the coefficients a, 6, c, d, &c., we have the following results : 



(2 2 - I 2 ) (3 2 - I 2 ) (4 2 - I 2 ) (5 2 - I 2 ) ... 

 2 2 .3 2 .4 2 .5 2 ... 



= A-BP 1 + CQi - DE, + ES i - &c., 



(I 2 _ 2 2 ) (3 2 -2 2 ) (4 2 -2 2 ) (5 2 -2 2 )... 

 1 2 .3*.4 2 .5 2 ... 



= A-BP,+ CQ. - DR 2 + ES 2 - &c., 



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





I 2 . 2 2 .4 2 .5 2 ... 



(1 _ 4) (2 2 - 4 2 ) (3 2 - 4 2 ) (5 2 - 4 2 ) . . . 

 I 2 .2 a .3 2 .o 2 ... 



= A - BP, + 4 - D^ 4 + ^^ 4 - &c., 



&c. 



Remarking the factors which are wanting to the numerators 

 and denominators to complete the double series of natural 

 numbers, we see that the fraction is reduced, in the first equation 



11 22 33 



to =- . o ; in the second to s T &amp;gt; m ^ ne third to - . ^ ; in the 



4 4 



fourth to -r . ^ ; so that the products which multiply a, 2&, 3c, 



4c, &c., are alternately ^ and It is only required then to 

 find the values of P&E&, P&R&, P 3 Q 3 ^ 3 ^ 3 , &c. 



To obtain them we may remark that we can make these 

 values depend upon the values of the quantities PQRST, &c., 

 which represent the different products which may be formed 



with the fractions ^ , ^&amp;gt; -&&amp;gt; T2&amp;gt; ^2&amp;gt; 7&&amp;gt; & c - without omit- 



1 L O TT O O 



ting any. 



With respect to the latter products, their values are given 

 by the series for the developments of the sine. We represent 

 then the series 



