PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 249 
with the like A-alues for d 2 y y, See. Substituting, the equation contains a term multiplied 
by a, viz. this is 
..Ja, 
which vanishes ; and a term multiplied by a, viz. this is 
i»((A) + (B)+(C)), 
which also vanishes. Writing down the remaining terms, and multiplying the whole 
by —V 3 , the equation becomes 
((A), . ,J a, . .) + ((A), . OCte, . .)-f 8 ((A), • • .JSX, 8Y, SZ) s =0. 
44. The last term admits of reduction ; from the equations 
(A)=— AV 2 +2XZW-2XWZ, &c., we find 
(A )&X + (H) IY + (G)&Z - - V 2 (A SX + HoY + GM)+ VA(Z oY - YhZ ), 
(H)5X + (B )IY + (F )IZ = - V a (H&X+B IY + FdZ) + VA(Xffi - Z XX), 
(G)$X+(F)&Y+(C )IZ = - V 2 (G SX + F XY + C hZ) + VXV(YXX - XSY ), 
and hence 
((A), . .pX, XY, IZ ) 2 = - V 2 (A, . . .£XX, XY, §Z) 2 ; 
wherefore the equation becomes 
((A), . 0& . .) + ((A), • • .) + 3(A, . .px, XY, XZ) 2 =0. 
45. It will be shown that we' have identically 
((A), . . .Ja, . . .)= — (A, . .px, JY, K5)*=2 
The partial differential equation thus assumes the form 
((A), . .Jta, • • -)+Q=o, 
where G may be expressed indifferently in the three forms, 
= +2(A, .), 
= + 2(A,..XiX, XY, XZ) 2 , 
= — 4 IX, XY, XZ 
X, Y, Z 
XX, XY, XZ 
46. Taking the first of these, the partial differential equation is 
((A), . . p«, . -) — 2((A), . .Ja, . . .)=0; 
XX, 
XY, 
XZ 
X, 
Y, 
z 
IX, 
SY, 
xz 
