246 PROFESSOR CAYLEY OY CURVATURE AYD ORTHOGOYAL SURFACES. 
37. We have 
(A") + (B") + (C")= (A" +B' +C")V 2 — (A", . .IX, Y, Z) 2 , 
A" + B" + C" =| { ( ZoY - Y8L)dj> + (X^Z - ZSXfe + ( YiX - XA )d z o } . 
Forming next the expressions of A'fX+IFY+GK'Z &c., and, for convenience, writing- 
down separately the terms which involve the second differential coefficients of §, we have 
A"X+H"Y + G"Z= 
d x p . V ( A 7j—cjY)-\- d tJ o\yi Z - ZoY +V(</X-«Z)]+ d z o[_ - (YoY -YA)-V (AX - «Y )}, 
H"X+B"Y + F"Z = 
^[-(V^Z-ZW)-Y(/Y-5Z)]+^.V(/X— AZ)+^[(ViX-XiV)+V(AY— iX)], 
G"X-fF"Y + C"Z = 
^[YW-YW+V(/Z- C Y)]+^[-(VSX-XW)-V(^Z-cX)]-F4?.V(^Y-/X), 
where oV stands for y (XSX+Y&Y -f-ZSZ), and where the three expressions contain also 
the following terms respectively : — 
{ . -YZd] + YZF + (Y 2 -Z 2 )II+ XY d z d x - XZ d x d y }q, 
{ ZXdl . -ZXd 2 - X Y d/L + (Z 2 — X 2 ) -f- YZd x d a }q, 
{-XY^ + XYdJ . + XZF/L- YZ^+(X 2 ~Y%^}g. 
Multiplying by X, Y, Z, and adding, the terms which contain the second differential 
coefficients disappear, and we obtain 
(A", . -IX, Y, Z) 2 = 2 Y[( Z£Y —YlZ)d x % + (X^Z — ZlX)d y % -f- ( YSX — XBY)c7^] ; 
so that, attending to the above value of M' + B'^C", we have the required equation 
(A")+(B")+(C")=0. 
38. Proceeding now to form the value of (A", . . •!(«), . . .), that is 
A" (a) + B r (l >) + C "(c) + 2F"(f) + 2G"(<?) + 2H"(A), 
it will be shown that the terms involving the first differential coefficients of g vanish of 
themselves ; as regards those containing the second differential coefficients, forming the 
auxiliary equations 
(A) =2(A)Z -2<y)Y, 
(B) =2(/)X-2(A)Z, 
(C) =2( ? )Y -2 (f)X, 
(F) =(/»)Y ~(g)Z -((J)-(e))X, 
(G) =(/)Z-(A)X -((«)-(»)) Y, 
(H) =(y)X-(/)Y-((a)-(S))Z, 
