Equations in Partial Differentials. 109 



Making x = {\y- + 2kY) + 2ky u, 



this, by the paper before referred to, reduces to 



(D2 + 2X-D + 2k'')u = x-' (D2 + 2kD + 2k'^)-i>{), 

 after restoring the suppressed factor and dividing by it. 



If l + ^^=a, 1--/^ = ^, 



we have D^ + 2ytD + 2k'^= (D + ak) (D + ^k) . 



But 



(D + uk)-p (D + g^)--PO=s-*^^ D-/'£2-'* ^~i D-z-e^''* 



Now, putting D' for k, and/ (?/) for A, &c., we have 



x-'(D + ak)-P(D + §k)-P0 = .v-U~'""^D-P{f{tj + 2x\^'^) 

 + .r/,(y + 2^v/31) . . . .+A'^'-'/p_,(2/ + 2.tV~f)} 



= a?-'e-«*"'-{(p(?/ + 2j??V"^) + ^^j(y-1.2AVZri) 



+ *?'-'<Pp_,(i/ + 2Tv/^) + .I,(3/)+.rvJ,,(y) . . . . 



+ af-'4'p_i(?/)} = £-«^*{a?-'<p(?/ + 2a7\/^) + ^, 



(j/ + 2^^^) ... . + a7''-2(Pp_,(?/ + 2a7^/~l)+a?-'iJ/(?/) 



where vj/, \{/„ &c. are introduced as the arbitraries of the inte- 

 gration. 



From the last result we easily see that 



+ «-'4/(_y) + &c.} = e-'"AD^-ie2^^-^^D''-'{«>-i^(j/) + &c. 

 + a;- ' vI/(2/ - 2j: \/^ ) + &c. } = e-«''-DP - 1 £-'^- ^~' {«-?' (p (3/) 



(_y + 2.r v/3i) + ^-P4,(y) + a^P+. 4,^(3/) .... +xP--^,p_,{j^) } 

 = r «'■''■{ a-^P+',p(^ + 2a? A/^)+.r-''^p+2,p,(y + 2^V3i) ... 



+ ^-''<p,._, {y + 2x\/^) +x-'P+'^7/)+x-^''+^4,i{7/) 



+ x-P^^,,_i{i/)}=x-^P+'{<p{i/-§x)+^i/-ax)}+x-^P+^ 

 {<Pi{t/-Sx)-{-^i{i/-ux)}+. . . . &c. 



In these reductions the values of a and S have sometimes 

 been put for these quantities, and k has been retained aa more 

 convenient than 1)'. 



As in the last examplo, by substitution in the proposed, we 



