SOLUTION OF THE DIFFERENTIAL RESOLVENT. 299 



Hence the general term may be written 



r (r— i)wj 



Now 



^ , , (r— i)7z 





^|y•^-m(/^— 1)| r ]w^ + ^^ — ^-\ 



2 '-i_P ^0^o,,„.+il^!ili=l^^(„.(„_.)+._rri)« 



TT 



whence we find the sum of the series 





^z\dxx''-' \ dO cos "«^^^i~^ 



cos ^ ^^ — cos'*^ cos J n—2 I Ox"^-^ 



n—i n'^ [ n—i J 



I S 1 cos^^ cos (/»— 2)^^^-'H ^^ ^- cos*'^^^*"-^^ 



By giving different values to (r), we obtain particular 

 integrals of the equation in succession. It is evident from 

 this investigation that the multiple integrals by which I 

 had previously expressed the solution of the differential 

 equation may be reduced to single integrals. In effecting 

 this we must^ of course, carefully restore the factors omitted 

 in the transformations given in this paper. 



Note. — To make this communication more complete, I 

 here insert the methods by which the series employed were 

 derived from the equations. The following rule to obtain 

 the series which express the solution of linear differential 

 equations when in the symbolical form, is extracted from 



