1885.] On certain Definite Integrals, 63 



Let P be the sum of the three cubes 



(\x + fxy + vz)* + (\'x + p'y + vzY + (\"x + // V + i>"z) 3 , 



which implies one condition between the constants a, b, and c. Then 

 we have 



r— r— r— 



f f [dxdydzx^y^h^CP) = 2/> — — - f 1 ^0(%)^ + ^" 1 - 



JJJ r(i+f+|) Jo 



where other things being the same, the limits are determined by 

 the equation 



Pt=L 



It is scarcely necessary to point out that these investigations may 

 be much extended when the radical sign is different, when the alge- 

 braical function under it is of a higher degree, and when there are 

 more than three variables. 



I enter now on a different subject. 



In a partial differential equation 



K4 %y=°> 



substitute for (u) a series of which the general term is Ax m y n . Then 

 if this series satisfies the equation, term by term, we have an alge- 

 braical equation f(m, n)=0, whence m=0(w), and the equation is 

 satisfied by an infinite series of the form 



Ax^ n )y n + BajM n i)y n i + Cx<t>( n *)y n * + . . . 



Now suppose y^ n ) to be expressed in the form fPQ n d<fi, then the series 

 is transformed into 



A/P(aQ)»<70 + B/T(a;Q)»i<Z0-|- . . . 



where A, B, <fcc., are arbitrary constants. 



Since the number of these constants is infinite, we may write this 

 (after Poisson) 



/PF(*Q)<Z0 



As an example, take the equation 



^ d^u d^u d»- l u _ Jb>u + jd v ~ l u , d v -*it, , 



*'d^d£ d^~ l di ^d^udg ' ' ~ a dy d^ 1 C 7^ z * 



