MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SURFACES. 
235 
Beside this we shall have for the two-branched contact at the fourth point, one 
equation in U of the degree 1, one of the degree 3, one in U and V of the degrees 1, 1, 
and one in V of the degree 1 ; so that the total number will be 
4 equations in U of the degree 1, (a) 
2 „ „ 3, (b) 
1 „ „ 9 , (c) 
6 ,, U and V of the degrees in U, 1 ; in V, 1, . . . (d) 
8 „ „ „ 1; „ 2, . . . (e) 
4 „ V of the degree 1 (f) 
Taking any five of the equations marked (d) and the four marked (f), we see that all 
the coefficients of the quadric surface are determinate and unique ; also that the quan- 
tities to which they are severally proportional are apparently of the degree 6 in the 
coefficients of U. Subject to future reduction of this number, the equations of condition 
in the coefficients of U and their degrees will be, 
4 of the degree 1, 
2 „ „ 3 , 
1 55 59 ^ 9 
1 „ „ 9, 
8 „ „ 13 . 
But having these ulterior reductions of degree in view, I at present abstain from 
enunciating the geometrical theorems which these results suggest. 
§ 2. General Theory of Multiple Contact of Quadrics with other Surfaces. 
The question may be considered from rather a more general point of view. Dropping, 
for the moment, the suffixes, and taking the symbol □ to represent any of the operations 
□ 12 , Du, . . (say □ ]2 ), let □' represent that part of □ which operates on U, and □" 
that part which operates on V; so that we may write symbolically □ = □'+□". It 
is first required to find the values of DU, D 2 U, .. in terms of □ , U, □ "U, . . The 
transformation will perhaps be better understood by examining two or three special 
instances before entering upon the general case. Thus 
□ U=D'U, v 
□ 2 u=n' 2 u+n''n'u, i (l) 
□ 3 u=n' 3 u+n"n' 2 u+ □'□"□'U+ □ ,,2 n'U.J 
But it will be found on developing the expression that □ ,,2 n'U=(12) 2 n , U; so that 
if, as is supposed in the present problem, the two surfaces touch at the point under 
consideration (a condition which is expressed by the equation □ , U=0), we have always 
MDCCCLXXVI. 2 K 
