708 PROF. W. K. CLIFFORD ON MR. SPOTTIS WO ODE’S CONTACT PROBLEMS. 
2. The surface u n is given, but not the point A. In this case, as the point A is only 
restricted to be a point on the surface, we have two more unknowns, making nine in all. 
The problems here are, to find the order of the curve on the surface at every point of 
which there is a conic having eight-point contact, and to find the number of points at 
which there is a conic having nine-point contact. 
As before, however, there are certain derived problems coming under this head which 
involve a less number of equations. We may seek the order of the curve traced out by 
points of contact of septactic conics satisfying one condition, sextactic satisfying two, &c. ; 
or we may seek the number of septactic conics satisfying two conditions, sextactic satis- 
fying three, &c. 
3. The surface is not wholly given. We may here assign a number of relations 
sufficient to eliminate the quantities a, b , c, leaving one or more relations among the 
coefficients of u n . The problems here are such as : — to find the number of surfaces in a 
pencil u n -\-w n which admit of ten-point contact with some conic, or one of whose nine- 
point conics meets a given line. 
In the present communication only problems of class 1 will be considered; the 
formulae in this case may be very considerably simplified. The quantities a and the 
coefficients of u n , then, are data of the problem ; so that the first of our equations, 
A”w„=0, is satisfied by hypothesis. The next equation, A n_1 BM re =0, signifies that the 
point B lies in the tangent plane at A, as it obviously must if the conic touch the surface 
at A. We shall suppose this also to be satisfied from the commencement ; that is, we 
shall regard B as a point moving in the tangent plane, and to be determined by con- 
struction in that plane. This may be effected analytically if we substitute for B, 
aA + ^B+vB', where now B, B' are regarded as fixed points in the tangent plane, and 
the three unknowns A, p, v take the place of the four quantities b. There is, however, as 
will be seen, no occasion to make the substitution explicitly. 
This being so, any relation involving B only beside the data must be regarded as the 
equation of a curve in the tangent plane. For example, A m-2 B^ n =0, expressing that 
B lies on the quadric polar of A, is the equation of the two chief tangents at that point. 
Generally, B"m„= 0 is the equation of the intersection with u n of the tangent plane; and 
the curves AB” -, w n =0, A 2 B re-2 w„=0, &c. are the successive polars of A in regard to that 
intersection. 
The terms entering into our equations are of the general form 
Qp+*i' r A. n - p - q WC q .u n . 
\n-p-q \p\q 
Any term, therefore, is completely determined by the two numbers p and y, and might, 
for any thing that has yet appeared, be denoted by (p, y). In view, however, of sub- 
sequent substitutions, we shall keep in evidence the manner in which B and C are 
involved, and denote the term in question by the symbol 6 p+2q (B P C ? ). 
As we do not consider any higher than seven-point contact, we have only the five 
equations : — 
