PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 707 
coefficient of Q m is the first that does not vanish, m roots of the equation coincide with 
zero, and m points of intersection are united at A. 
The result of substituting the coordinates of any point P in u n may be conveniently 
represented by means of the differential operator P. It is known, in fact, that 
(*» % 3 , x,) n ^\ n. (p lt p„ p 3 , Ihf ; 
or, which is the same thing, P n u n is | n times the result of substituting the coordinates 
of P in u n . Our result may therefore be stated in the following form : — 
The necessary and sufficient conditions that the conic K 2 = 4AC— B 2 may have m -point 
contact with the surface u M at the point A, are that in the expansion o/(A+0B+0 2 C) n .u„ 
in powers of 6, the rath power of (l is the lowest whose coefficient does not vanish. 
II. 
Equating to zero the coefficients of 1, 0, S 1 in this expansion, we obtain 
0=A”. u n , 
0=mA” -1 B . u n , 
0 =.\n{ii — I) A” _2 B 2 . u n +nA. n - l C . u n . 
Before proceeding further with these equations, it is convenient to make the following 
remarks upon their nature, which will serve to simplify the expression of them. 
In the first place, then, we have here a series of relations among the coordinates of 
the points A, B, C and the coefficients of the surface u n ; and the determination of the 
coordinates and coefficients so as to satisfy a certain number of the relations presents us 
with the solution of various geometrical problems. These problems fall naturally into 
three classes. 
1. The surface u n and the point A are given. In this case the unknowns are the 
ratios of the eight quantities h, c, a singly infinite number of solutions corresponding 
to each conic; and we are accordingly able to satisfy seven of the equations*. The 
problem here is to find the number of conics which have seven-point contact at a given 
point of a given surface. 
We may, however, impose beforehand certain restrictions upon the values of the 
unknowns, and so consider problems which involve a less number of the equations. 
While the number of the septactic conics is definite, the sextactic conics form a singly 
infinite series ; and we may ask what is the number of them : 
(a) whose planes pass through a given point, 
( b ) which meet a given line, or 
(c) which touch a given plane. 
The quintactic conics, again, form a doubly infinite system, and we may inquire about 
the number of them which satisfy two conditions ; e. g. which pass through a given point. 
* Yiz. six besides the first, which is satisfied identically. 
5 c 
MDCCCLXXIV. 
