714 PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE'S CONTACT PROBLEMS. 
If, now, we wish to find the nature of the curve in which this quadric 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 which is of the order n in 0 and in <p 
separately. If we regard 0 and <f> as coordinates of a point on the quadric surface, the 
equation just found is that of the curve of intersection. 
Suppose that in this equation the term independent of 0 and <p vanishes, then the 
equation is satisfied by the pair of values 0=0, <p=0, or the curve of intersection passes 
through the point A. Now the various directions in which we may start from the point 
A are determined by the initial value of the ratio 0 : <p when we move along them. The 
direction in which the intersection-curve starts from A is therefore that obtained by 
neglecting in the equation terms of higher order than the first ; and we see that there is 
only one such direction. 
If, however, not only the constant term but the coefficients of & and <p in the equation 
vanish, the initial directions are obtained by equating to zero the terms of the second 
order, i. e. by neglecting in the equation all terms of higher order than the second. In 
this case, then, there are two such directions, the intersection-curve has a double point 
at A, and the quadric has with the surface u n an ordinary or two-branch contact. 
Again, if the coefficients of the terms of the third order are the first that do not 
vanish, the initial directions are obtained by neglecting all the terms of higher order, 
and there are consequently three of them. Thus the intersection-curve has a triple 
point at A, and the two surfaces have a three-branch contact. 
And so generally, if the coefficients of the terms of the with order are the first that 
do not vanish, the intersection-curve has at A a multiple point of order m, and the two 
surfaces have at that point an m -branch contact. 
The result of these considerations may be stated as follows 
The necessary and sufficient conditions that the quadric Q 2 =AD— BC may have 
m-branch contact with the surface u„ at the point A, are that in the expansion of 
(A+ 0B + <pC-f- 0<pD)”. u„ in powers and products of Q and <p, the terms of order m in 0 
.and <p are the lowest whose coefficients do not vanish. 
II. Quadrics of four-branch contact. 
The equations we shall have to employ are so simple in form that it is unnecessary to 
employ the abridged notation of the former Part. We shall merely omit the operand 
u n , and any common factor of the binomial coefficients. 
The conditions for ordinary contact are then 
0=A", 0=A ra_1 B, 0=A” -1 C. 
The first of these expresses that A is a point on the surface u n , the second and third 
that B and C are on the tangent plane at A. 
The further conditions for three-branch contact are 
0=A"- 2 B 2 , 0=A”- 2 C 2 , 0 = A” -1 D + (n — 1) A” _2 BC. 
