and their Index Symbol. 139 



(II.) xp + yq-^~^.z = 0, 



m 



z=u (l + ® n ) n . 



11. Again, it is shown in the same memoir that the solution 

 of ordinary differential equations of the class 



u + af(D) .e^ M + i/(D)./(D-m).e2»»e M + &c. =0, 



is reducible to the solution of the system of equations 



u-q 2 f(D)e^ u =0 L 

 &c. J 



where q v q v &c. are the roots of the equation 

 q n + aq n - 1 + bq"- 2 + &c. =0. 



Similarly, the solution of the class of partial differential equations 

 represented by 



2 + A.F(V).© m £ + B.F(V).F(V-M).©L?-f&c. = 0, 

 is reduced to the solution of the system 



z- Q,.F(V). ®,^=0- 

 z- Q 2 .F(V).© ro *=:0 

 &c. 



where Q v Q 2 , &c. are the roots of the equation 

 Cr + AQ' ! - 1 +BQ M - 2 + &c. =0. 

 It is obvious that partial differential equations of the class 

 * + A© m .F(V)* + B©;,.F(V + »O.F(V)* + &c. =0 

 admit of a similar reduction. 



12. It remains to examine the application of the general prin- 

 ciples to the subject of multiple definite integrals, an application 

 of which they are obviously susceptible, and which seems to open 

 an interesting field for investigation. It would be impossible, 

 however, here adequately to follow up such an inquiry in its 

 details, and a general theorem, with its application to two par- 

 ticular cases, must for the present suffice. 



If 



rdxfdyJdz...Q.{xyzkc.).a^ x y z8i ^ .b x ^ z&c ^ .e^* &c -)... = K, 



the quantities «, b, c, &c. being unconnected with the limits, 

 then will 



f,h fdy/kz.M(aygkc.).Y(fl>+x+1r+bc.).rf. /;*.cf..=F(n).K 



