40 
C. VIII. 
519] ON CURVATURE AND ORTHOGONAL SURFACES. 313 
44. The last term admits of reduction ; from the equations 
(A) = — AV 2 -\- 2XZS Y - 2XYSZ, & c , we find 
(A ) SX + (H) 8Y + (G) SZ = — V 2 (ASX + H8Y + GSZ) + VSV(ZSY - YSZ ), 
(H) SX + (B)8Y+ (F) SZ = — V 2 (USX + BSY + FSZ) + VSV(XSZ - ZSX ), 
(G) SX + (F) 8Y + (C) SZ = - F 2 (£SX + FSF + GSZ) + V8V(Y8X — X8Y), 
and hence 
((A),.J8X, 8Y, 8Z) 2 = — V 2 (A, ...$8X, SY, SZ) 2 ; 
wherefore the equation becomes 
((il),..$«,..)+ (W,..$8a..) + 3(A,..pz, 8Y, 8Zy = 0. 
45. It will be shown that we have identically 
((4), ...) = - (A,..^SX, 8Y, SZ) 2 = 2 8X, 8Y, 8Z 
X, Y, Z 
EX, SY, SZ 
The partial differential equation thus assumes the form 
((¿),..pa,...) + il = 0, 
where CL may be expressed indifferently in the three forms, 
= + 2 (A,..][«,..), 
= + 2(4,..%8X, 8Y, SZ) 2 , 
= — 4 SX, SY, SZ . 
I X, Y, Z 
EX, EY, SZ 
46. Taking the first of these, the partial differential equation is 
((A), ...$Sa,..)-2((4)„.$5,...) = 0; 
or, written at full length, it is 
(A) Sa + (B) 8b + (G) Sc+2 (F) 8f+ 2 (G) 8g + 2 (H) Sh 
-2 {(A) d + (B) b + (G) c + 2 (F) f +2 (G) g + 2(H)h} = 0, 
where the coefficients are given functions of X, Y, Z, a, b, c, f g, h, the first and 
second differential coefficients of r; and 8 is written to denote Xd x + Yd y 4- Zd z .