228 
ME. W. SPOTTISW 0 ODE ON MULTIPLE CONTACT OE SUEFACES. 
§ 1. 'Recapitulation of former Methods and Results. 
Let x, y,z,t; x^y^z^, t x ; . . be the coordinates of the points P, P,, . . respectively ; and let 
U=(x,y,z,t) n =0, V=(x,y,z,t) m =0, (1) 
be the equations of the two surfaces whose contact is the subject of consideration. The 
conditions of contact may, as is well known, be written thus : 
ajj : a y u : a,u : a jj i 
=a,v : : a 2 y : a,v. j 
Another form of these conditions is, however, better adapted to our present purpose. 
In fact, writing 
d y vd 2 u-d s vd y u=& u, a^a/u-^yajj^u, j 
a 2 va :r u-a,va 2 u=^ 1 u, a y va,u-a,ya,u=^ 4 u, l (8) 
a I va y u-a y va 3: u=^u, a 2 va < u-a,ya 2 u=^u, J 
or, more briefly, 
ajj, a y u, a z u, ajj 
a x v, a y v, a s y, a,y 
=&U, . . S 5 U, 
(4) 
we may take as the conditions necessary, in order that y shall touch U at the point P, 
any two of the six following, viz. : 
SU=0, ^U=0, . . UJ=0, (5) 
Similarly, as I have shown in a memoir “ On the Contact of Surfaces ” (Phil. Trans. 
1872, p. 259), w 7 e may take as the further conditions that y (assumed to touch U at the 
point P) may have a three-branch contact* with Y at the point P, any three inde- 
pendent equations of the following system : 
m=o, vu=o, . . ^u=o, (6) 
and so on for higher degrees of contact. 
For the purpose of the present inquiry it will be convenient to transform the equa- 
tions (5) and (6) into yet another shape. Thus, if we write 
(x,y,z,t) n = 0 n , (x,y,z,t) n ~ 1 (x 1 ,y 1 ,z 1 ,t 1 )=O n 'l,] 
(x,y,z,t) 2 = 0 2 , (x, y, z, t) (ff„y„3i, *i)=01, J 
and multiply each number of the system (2), first by x x ,y„z 1 ,t 1 respectively, and add, 
then by x 2 , y 2 , z 2 , t 2 , and add, and so on, we shall obtain the systems 
0""T : O-^ : . . 1 
= 01 : 02 : . . J 
( 8 ) 
* The terms 2-branch, 3-branch, &c. contact, already used by Professors Cayley and Ceiffoed, have the 
following signification : — in ordinary contact the curve of intersection has at the point a double point, or two 
branches ; in contact of the second order, a triple point, or three branches, and so on. 
