300 
ON CURVATURE AND ORTHOGONAL SURFACES. 
[519 
We have thus the system of equations 
(A) = -AV 2 . +2XZ8Y -2 XY8Z, 
(B) = - BV n - - 2YZ8X . + 2XY8Z, 
(G) = -CV 2 +2 YZ8X - 2XZ8 Y 
(F) = - FV 2 + (Y 2 - Z 2 ) 8X - XY8Y + XZ8Z, 
(G) = -GV 2 + XY8X + (Z 2 -X 2 )8Y- YZ8Z, 
(H) = -HV 2 - XZ8X + YZ8Y +(X 2 -Y 2 )8Z. 
20. We hence find 
(.4) a + (H) h + (G)g = - (Aa + Hh + Gg) V 2 + (Z8Y - Y8Z) 8X + XP, 
(H) h + (B)b+ (F)/= - (Hh + Bb + Ff) V 2 + (X8Z - Z8X )8Y+ YQ, 
(G) g + (F)f +(C)c=- (Gg + Ff +Cc) V 2 + (Y8X - X8Y)8Z + ZB, 
if for shortness 
P=( g Y- hZ) 8X + (aZ - gX) 8Y+ (hX - aY) 8Z, 
Q = (fY- bZ)8X + (hZ-/X)8Y+(bX -hY)8Z, 
R = (cY -fZ) 8X + (gZ - cX)8Y + (fX-gY) 8Z. 
Forming the sum PX + QY + RZ, the coefficient of 8X is found to be 
= - Z(hX + bY+fZ) + Y (gX +/Y + cZ), =-Z8Y+Y8Z- 
hence the whole is 
= 8X(Y8Z-Z8Y) + 8Y(Z8X-X8Z) + 8Z(X8Y- Y8X), which is =0, that is, 
PX + QY+RZ = 0. 
21. Hence, adding, we find 
viz. in this and the before-mentioned equation 
(4) + (P)+(C) = 0 
we have the a posteriori verification that the cone 77, £) 2 = 0 cuts the tangent 
plane in the double lines of the involution. 
In what precedes I have given only those relations between the several sets of 
quantities a, a, (a), A, (A), &c. which have been required for establishing the results 
last obtained; but there are various other relations required in the sequel, and which 
will be obtained as they are wanted.