310 
ON CURVATURE AND ORTHOGONAL SURFACES. 
[519 
Multiplying by X, Y, Z, and adding, the terms which contain the second differential 
coefficients disappear, and we obtain 
(A",.JX, Y, Zy = 2V[(Z8Y-Y8Z)d x p + (X8Z-Z8X)d y p + (Y8X-X8Y)d z p]-, 
so that, attending to the above value of A" +B" + G", we have the required equation 
(A") + (B") + (G") = 0. 
38. Proceeding now to form the value of {A", ...), that is, 
A" (a) + B" (b) + G" (c) + 2F" (/) + 2G” (g) + 2H" (h), 
it will be shown that the terms involving the first differential coefficients of p vanish 
of themselves ; as regards those containing the second differential coefficients, forming 
the auxiliary equations 
(A) = 2(h)Z —2(g) Y, 
(B) =2(f)X-2(h)Z, 
(G)=2(g)Y -2(f)X, 
(F) = (h) Y — (g) Z — ((6) — (c)) X, 
(G) = (f) Z — (h) X- ((c) -(a)) Y, 
(H) = (g)X- (f) Y — ((a) — (6)) Z, 
we find without difficulty that the terms in question (being, in fact, the complete 
value of the expression) are 
^((M), ..Jfrd x , dy, d z y p. 
39. As regards the terms involving the first differential coefficients, observe that 
the whole coefficient of d x p is 
(YHY-Z8Z)) 
+ 2 
which is 
= 2V{(g)h + (f)b + (c)g- ((h)g + (b)f+ (/) c)} 
+ y{2 ((h) BX + (b) 8F+ (/) 8 Z) - Y ((g) SX + (f) SY+ (c) SZ)}.