26 The Rev. S. Earnshaw on the Transformation 



and this by writii)g a'a;^ —Uy for .r and y ; and assuming 

 C« = fl'Z;'D, takes the form (1.); to which, therefore, the general 

 equation proposed can always be reduced when C^ is not 



2. But when C^ = 4-ai, then as we cannot make the above 

 assumptions for^ and g', we may suppose ^ = 0, and a + Cg' 

 4-^a'2=0; it will then be found that the third term of the first 

 transformed equation in art. (1.) will vanish of itself, and thus 

 the transformed equation is of the form 



If we proceed with this as with (5.) it becomes 



and as /, m are arbitrary, we can always make the second and 

 last terms of this to disappear ; by which means it takes the 

 form 



d^u „, du 



which may be further reduced, as before, to the form (2.). 



3. To transform 



d^u J d^u d'^u . d^u ^ d^u p d'^u . , du 

 dx^ dy'^ dz^ dydz dxdz dxdy dx 



„, du ^, du „ 

 + B'^+C'^ +Dm=0, 



the general partial differential equation of the second order 

 with three independent variables. 

 Change the variables by the assumptions 



^ = x+gy, ri=x+g'y, ^=x-^hy^kz. 



This gives 



0=^a^bg^+Cg)^ +{a + bg^^ + Cg')^ +{a + bm + ck^ 



+ AA^ + B^ + C^) ^ + (2a + Cy + 2^?+C.//+ A^^+B./f ) 



|^ + (2« + Cg+26^TC.A+A^B.^)0^ 



d^u , . , ^, ^ du 



+ (2a + C.g+5'4-2^'gg')^P^- +(A' + B'^)^^ 

 + (A'+B'^')^' +(A' + B'/% + C'/t)^' +D«. 



