348] 
ON THE THEORY OF INVOLUTION. 
309 
40. If the two conics are 
(a, b, c,f, g, hjx, y, zf = 0, 
(a', V, c', f, rj, h'Jx, y, zf = 0, 
and if the determinant formed with a+ka', &c., is denoted by 
(K, ©, ©', KJ1, k)\ 
so that 
K = abc — af 2 — by 2 — ch 2 + 2fgli, 
© = a' (be — / 2 ) + &c., 
©' = a (6V — / /2 ) + &c., 
K =a'b'c' -cl'/' 2 — b'g' 2 — c'h' 2 + 2f'g'h', 
then the before-mentioned equation □ = R = 0 which gives the condition that the 
conics may touch is 
Disc*. {K, ©, ©', K’\ 1, lcf = Q, 
where the left-hand side is of the order 6 in the coefficients of the two conics 
respectively: this is a known formula. 
41. If the equation in k have a twofold root the two conics will touch: two of 
the critic centres will then coincide at the point of contact, or this point is a 
twofold critic centre: the remaining or onefold critic centre is the intersection of the 
common tangent and of the line joining the two points of intersection of the conics. 
In virtue of the general property, the first-mentioned two centres must be considered as 
lying on the line which is the polar of the onefold critic centre in regard to either of 
the conics. The diacritics pass through the critic centres, that is, they pass through 
the onefold centre, and touch the polar in question at the point of contact of the 
two conics, or twofold critic centre; this is in fact the property mentioned ante, No. 28. 
42. The equation 
Disc*. Disc*. (U + kV) = 0, 
as applied to a conic and a circle leads at once to the equation of the curve parallel 
to a given conic; such parallel curve is in fact the envelope of the circles of a given 
constant radius which touch the given conic. This method is in effect due to 
Dr Salmon, who applied the corresponding theorem in solido to the determination of 
the surface parallel to an ellipsoid. 
Annex, referred to No. 31. Investigation of the order of the plexus or system 
for the existence of a Cusp. 
Considering for a moment the curve 
U = (*$æ, y, z) n = 0, let (L, M, N), {a, b, c, f g, h) 
he the first and second differential coefficients of U; (A, B, C, F, G, H) the inverse 
system, viz. A = be — / 2 , &c. At a cusp we have 
1 = 0, M = 0, N= 0, A = 0, B = 0, (7 = 0, F = 0, (7 = 0, H= 0,