706 PEOF. W. K. CLIFFORD ON ME. SPOTTIS WO ODE’S CONTACT PEOBLEMS. 
Part I.— THE CONTACT OF CONICS WITH SURFACES OF ORDER n . 
I. 
The current plane-coordinates being denoted by 
X„ X 2 , X 3 , X 4 , 
let the equations of the three points A, B, C be respectively 
0 == ttjXj -J- $ 2 X 2 “l - ft 3 X 3 -f - £jqX 4 = 2® X , 
0 =2&x, 
0 = 2cX. 
The quantities a { , c { (i= 1, 2, 3, 4) are the coordinates of the points A, B, C. The 
symbol A itself I shall use indifferently, as denoting either the form 2 «X or the differ- 
ential operator 
0 A, + «A 2 + «A 3 + «A, = SoA, 
where # 2 , # 3 , # 4 are the current point-coordinates. It will be seen in the sequel that 
this double meaning is useful, while it does not introduce any confusion. Similar 
interpretations are of course to be given to the symbols B, C, and the like. 
Consider now the point 
P=A + 0B-f-0 2 C, 
whose coordinates are a t -|- 0^ + = 1 , 2, 3, 4). If we suppose 0 to take all possible 
values, the point P will describe a conic section whose tangential equation is 
0 = 4AC— B 2 —K a . 
To the value 0=0 corresponds the point A, to 0=co the point C; while the equation 
shows that B is the intersection of the tangents at A and C. 
To find the point at which this conic intersects a given surface u n of the order n, we 
must substitute the coordinates of P in the equation of the surface ; in this way we shall 
form an equation in 0 of the order 2 n, the solution of which will give the values of 0 
belonging to the 2 n points of intersection. 
If in this equation the term independent of 6 vanishes, then 0=0 is a root of the 
equation ; consequently the point A is one of the intersections, or the surface u n passes 
through the point A. If also the coefficient of $ vanishes, another root of the equation 
coincides with zero, and two points of intersection are at A. And generally if the 
