Equations in Partial Differentials. J 1 1 



Substituting this value in the given equation, vi^e find 



;r=^P+' |^(j/) + vf,(V + ^)]>+ .r'' |«<p'(3/)+^rI.'(i/ + ^)]> + 



&,c., the series in (p, <^\ &c. stopping at .r°, that in vf/, i)', &c. at 

 x^l the values of the constants a, a^, &c., b, b^, Sec. are not 

 put down, for the same reason as in the last example. 



If in this last example we make ^ = PD' + Qj P and Q 

 being functions of y, the equation to be integrated will be 



which by z = D''~^u reduces to 



^' ^ + P ^+(Q-2p^-)«=/(.y) +.^/i(^) . • . .+^-^->/p-.(j/), 



which is only of the first order. We must not proceed in this 

 case as heretofore, but must expand 



by the powers of- —, if we wish to integrate as in the 



preceding examples. Thus 



/(i/)=/(y)+*-'(PD' + Q)/(i/)+f"(PD'+Q)(PD'+Q) 



/(j/) + &c., 

 the powers, after the expansion, being changed into successive 

 operations. In this case we might not obtain a solution in 

 finite terms. 



Sometimes it may be desirable to proceed with D' in the 

 same manner as with D. Let 



or xI>^z + xDD'is + Dz = 0. 



But, as in the papers referred to, 



xD^z = D^xx-2l>z, xDB'z = DD'x::-jy^. 

 Therefore D^xz+BD'xz-Dj:-B'z=0; 



or (D + D')(D.r.r-^) = 0; 



or 



Dx^-Z:=(D4-D')-'0 = e-^"' D-'r'i''0 = e-''^7(i/) =/(?/-. r) ; 



PD' 



