ME. W. SPOTTISWOODE ON THE CONTACT OF SURFACES. 
277 
identically. Similar remarks apply to contact of the degrees 6, 8, 10 ; but not appa- 
rently to that of the degree 15. 
It will be worth while to examine the case of four-pointic contact more in detail ; 
and for this purpose the special transformation of the operation □ 2 V, employed in the 
memoir before quoted, appears to be best adapted. The following method of effecting 
that transformation is perhaps more expeditious and direct than the process used in the 
memoir itself. 
Taking for □ the form Aoj + BT-j-Coj, we have 
□ 2 v=n 
A, 
u , 
— 
<! 
n 
u. 
+ 
A, 
□ n , 9,, 
+ 
A, u , 
□ B x3 
B, 
v, 
□ B, 
v, 
** 
B, 
□ v , h !/; 
B, v, 
□ B* 
c, 
w. 
3* 
□ c, 
W, 
c, 
□ w, 
C , tv, 
□ B g . 
But by the equations (11) of the memoir, 
A, 
u , 
aA—aA, 

A, 
u, 
a, 
A= ZU 
a, 
a', 
u. 
B, 
v. 
«B — j3A, 
B, 
V, 
P, 
/3, 
P, 
v. 
c, 
IV, 
aC — yA, 
c, 
IV, 
r/ 
/ ? 
Vv 
y\ 
w, 
whence 
□ A, 
u , 
B*, 
Y= — 
a, 
a! , 
u , 
A, 
u , 
Bj;5 
Y= — 
a, a! , 
u, 
□ B, 
V, 
ft 
V, 
V, 
B, 
v , 
By, 
(3, (37 
v , 
□ c, 
IV, 
B„ 
Y> 
w, j 
c. 
tv. 
B* 
7-> r /> 
w. 
since by hypothesis □Y=0. Again, 
□ «= 
A, 
u , 
, □«= 
A, 
u , tv'. 
, Qtv — 
A, 
u, 
v' , 
B, 
V, 
tv', 
B, 
V, V,, 
B, 
v. 
u' , 
c. 
tv. 
v' , 
c, 
tv, u 
c, 
tv. 
w:, 
whence, remembering that (on the supposition □Y = 0) ‘d x Y=Qu, d.V =0 W , 
we derive the following : 
A, 
nu. 
B„ 
II 
A, 
□ «, 
W, 
= -0 
U x , 
2t/, 
v' , 
it , A, 
B, 
nv , 
B, 
□ v , 
tv', 
v x , 
v! , 
v , B, 
c, 
Uw, 
B„ 
c, 
□ 20, 
w, 
v ' , 
it! , 
tv i, 
tv, C, 
u , 
v , 
tv, 
• 
A, 
B, 
c. 
. 
2 p 2 
