42 
ME. W. SPOTTISWOODE ON THE CONTACT OF CEEVES. 
so that, writing 
/ BU ^BU\ B . / ^ — =□ 
\yi:^-^~d:^)d:x^y dz-"^ dx) dx dy)dz 
. (6.) 
the effect of the operation □ upon V is equivalent to the elimination of dx, dy, dz, 
from (1.) and their differentials (2.), and is expressed by the equation 
0 -) 
If the curves have three consecutive points in common, we have, in addition to former 
conditions. 
from which dx, dy, dz, d% d% d^z are to be eliminated by the help of (1.) and 
(2.), or (3,). Instead, however, of differentiating (2.), we may differentiate (3.), or, 
what is the same thing, (4.); and we shall then have an expression free from d?x, 
d'-y, cVz. 
The results of the elimination may, moreover, in this case be grouped into a single 
expression. Writing for convenience 
BU B BU ^ -i-x BU ^ BU ^_p) 
Jz~ dz dij~~^^'‘ dz dx dx dz 
^ B 'd^jy 
dx dy dy dx ■ 
(9.) 
(3.) may be expressed by any two of the equations 
D,V=0, D,V=0, D3V=0 (10.) 
Differentiating these and combining their differentials with the first of (2.), we may form 
the quadi’atic system. 
D?V =0 
D,D,V=0 
D.D3V=0 
D,DiV=0 
D|V =0 
D3D3V=0 
(11.) 
D3DiV=0 
D3D3V = 0 
D'V =0, j 
all of which are comprised in the 
one equation 
□w=o 
And generally, by a similar train of reasoning, the constants of a curve V, having an 
m-pointic contact with U, will be determined by the equations 
V=0, □V=0, .. □W=0; (13.) 
and if 
Yz={a,b,.. y, zf 
(14.) 
