337 
411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
and representing this for a moment by 
A, 
H, 
G, 
L 
= 0, 
H, 
B, 
F, 
M 
G, 
F, 
G, 
F 
L, 
M, 
F, 
D 
then in the developed equation 
D (ABC - AF 2 - BG 2 - CH 2 + 2FGH) 
— (BC — F 2 , CA — G 2 , AB - H 2 , GH-AF, HF - BG, FG - CHQL, M, N) 2 = 0, 
observing that C, F, M, JSf, D are of the first order in x, y, the only terms of the 
first order are contained in B (- DG 2 - CL 2 + 2NGL); and since C, D, N are of the 
first order, we obtain all the terms of the first order by reducing B, G, L to the 
values 2-iJr, C, D; viz. the terms of the first order are 
2yjr (- C 2 dx - D-cx + 2CDnx), = -2f (C 2 d + D 2 c - 2CDn) x. 
Hence the complete equation is of the form 
— 2-v^ (C 2 d + D 2 c — 2CDn) x + (x, y) 2 = 0, 
or, what is the same thing, x^> + y 2 T = 0; the Hessian has therefore along the line 
x — 0, y = 0 the same tangent plane x = 0 as the surface; or it touches the surface 
along this line; that is, the line counts twice in the intersection of the two surfaces. 
28. If instead of the right line we have a plane curve, say if the equation be 
x<f> + P 2 yfr = 0, then the value of the Hessian is + Pfi' - = 0 (viz. the second term 
divides by P only, not by P 2 ), so that, as before mentioned in regard to a conic of 
contact, the surface and the Hessian merely cut (but do not touch) along the curve 
x = 0, P = 0. To show this in the most simple manner take the equation to be 
xcf) + hP 2 = 0; let A', B', C\ P>' be the first derived functions of 0, and (A, B, C, D), 
(a, h, c, d, f, g, h, l, m, n) the first and second derived functions of P; then if in the 
equation of the Hessian we write for greater simplicity x = 0, the equation is 
2A' + Pa+ A 2 , 
B' + Ph + AB, 
C' + Pg + AC, 
D' + Pl + AD, 
B' + Ph + AB, 
Pb +B 2 , 
Pf +BC, 
Pm+ BD, 
C' + Pg + AC, 
Pf + BC, 
Pc +C 2 , 
Pn + CD, 
D' + Pl +AD 
Pm + BD 
Pn + CD 
Pd + D 2 
= 0. 
The equation contains for example the term 
- (U + PI + AD) 2 {P 2 (be -f 2 ) + P (bC 2 + cB 2 - 2/BC)}, 
dividing as it should do by P, but not dividing by P 2 ; and considering the portion 
hereof — D' 2 P (bC 2 + cB 2 — 2/BC), there are no other terms in D' 2 P which can destroy 
this, and to make the whole equation divide by P 2 ; which proves the required negative. 
O. vi. 43