348] 
ON THE THEORY OF INVOLUTION. 
305 
that is 
x : y : z = bc -f 2 : fg - ch : hf-bg 
—fg — ch : ca — g- : gh — af 
= hf — bg : gh — af : ah — h 2 , 
so that the above written equation is Lx + My + Xz = 0, which is true in virtue of the 
equation ¿7=0; and similarly for all the other equations which were to be verified. 
28. It is to be noticed that the determination of the tangent of the diacritics 
depends only on the second differential coefficients (a, b, c, f g, li), (a', b', c', /', g', li'), 
of U, V. The tangent in question will be the same if instead of the curves, ¿7 = 0, 
V = 0 we have the conics (a, b, c, f, g, }i$x, y, zf = 0, (a, b', c', f, g', h'][x, y, zf = 0: 
these conics pass through the point of contact of the two curves, and their tangents 
are coincident with those of the two curves ¿7=0, V = 0 respectively; that is, the 
conics touch at the point in question. They consequently intersect in two more points; 
the chord of intersection or line joining the last-mentioned two points, meets the 
common tangent in a point; the polars of this point in regard to the two conics 
respectively, pass through the point of contact, and moreover they are one and the 
same line; this line is the required tangent of the diacritics. The proof will be 
given, post, No. 41. 
29. Let R = 0 be the condition in order that the two curves ¿7=0, V= 0 may 
touch each other, or say let R be the Tactinvariant of ¿7, V. When the curves 
¿7, V are of the degrees m, n respectively, then R is of the degrees n (2m + n — 3), 
m (m + 2n — 3) in regard to the coefficients of ¿7, V respectively. Hence in the 
present case where ¿7, V are each of the degree n, R is of the degree 3n (n — 1) in 
regard to each set of coefficients. 
30. Secondly, if the functions U, V are such that there exists a curve U + kV= 0 
(say the curve ¿7 +&jF=0) which has a cusp, then it is to be shown that k = k ± is 
a twofold root of the equation in k; and to do this it has to be shown that the 
cusp is a twofold critic centre; or that the diacritic curves touch at the cusp: it 
may be added that the cuspidal tangent is the common tangent of the diacritic 
curves. Now the cuspidal curve being U+k 1 V=0, then at the cusp the first derived 
functions L + J^L', M + ^M', N + kiN' vanish identically; and moreover the second 
derived functions a + ka',... are such that (X, Y, Z) being any magnitudes whatever, 
(a + k 1 a',...'$[X, Y, Zf is a perfect square, = (\X + gY + vZf suppose. Now X, Y, Z 
being current coordinates, and D denoting the operation D = X8 x +Y8 y +Z8 z , the 
equation of the tangent to the diacritic curve (by an investigation similar to that for 
this same tangent in the case first above considered) is found to be 
L + dL' 
a 
s 
M + dM' 
x+ex' 
7 
= 0, 
C. Y. 
39