2-12 PXOFESSOE CAYLEY OX CUEYATUEE AND OETHOGONAL SURFACES. 
say these are A'=A-fQA, &c., where 0 is a functional symbol; we thus have 
(A', . . ,I|', ?7=(A, . . .If, ?)" + (« A, . . .If, ,, ?)> + 2(A, . . .If, „, ?IAf, Aij, A?), 
which, for shortness, I represent by 
=(A,..0CI, * £) 2 +(A", . . * £) 2 ; 
and I proceed to complete the calculation of the coefficients A", B", &c. 
30. We have 
A'r^A-f-coeffi | 2 in 
2[(A^+H, + GOA|4-(H§+B 9 +F?)A?+(G|+F^+C?)Aa 
= 0A + 2 (A djgx . A H djgfi A Gd x jy), 
that is 
A" = 0 A + 2 y(AX A HY A G/)d t o 
+ 2^( A d x a A YLdJfr A Gd//), 
where coeff. 2^ is 
Art + H h + G g (AX + H Y + GZ)8X 
Y _ V 3 
=\^{hZ -gY) + AZ -#Y j - ^(ZiY - Y&Z). 
31. And similarly, 
F" = 0F A (Ha A Bj3 A Fy)^ 2 g> A(G«AF|3A C y)d y % 
A § { ( H dsi A B djfi A F ds / ) A ( G d y c& A F d y fi A C d y y ) } 
= 0F A \ { HX A BY A FZ)Ag>A (GX A FY A CZ)d^} 
(% + B/+ Fc 
(HX + BY + FZ)8Z 
q v 
Y 3 
Gh + Fb + Cf 
(GX + FY + CZ)8Y) 
+ y 
Y 3 b 
Gh A FA A C f= o(hY - AX) A AY -bX+cuX + aX A AY +gZ, 
IF/ A B/A F c= -v(gZ-cX)-gY + cX-vX-aX-kY-gZ. 
Sum is &>{ AY —gZ — {b — c)X } A AY —gZ — (b — c)X, which is = coF A AY —gZ — (b — c)X 
F'=ff+(XSZ-Z8X)(ii,g-?||t +(YSX-XSY) (A»§- 
gSY 
\3 
hence 
