278 
where 
ME. TV. SPOTTISWOODE ON THE CONTACT OE SUBFACES. 
w' , 
v' , 
l' , 
A, 
w', 
v r 
u n 
u ' 
to', 
B, 
v' , 
it ! , 
'll! , 
c, 
l' , 
to', 
it ! , 
hi 
B, 
A, 
B, 
c, 
B, 
• 
Again, if in the following expressions ~d x , ch, d ( be understood to affect V alone, then 
A, 
u , 
□a,, 
V= 
w', 
v' , 
A, 
a— 1 
w'. 
^ , 
? , 
A, 
a,, 
B, 
^ , 
□a„ 
w', 
®i, 
u', 
B, 
w', 
«i» 
u’ , 
to', 
B, 
3,. 
c, 
w, 
□ a*, 
w', 
W„ 
w, 
c, 
v' , 
w', 
Wm 
u’ , 
c, 
a. 
** , 
V » 
W, 
• 
. 
. 
u , 
0 , 
W , 
k , 
• 
. 
A, 
B, 
c, 
• 
• 
• 
A, 
B, 
c, 
B, 
. 
. 
a„ 
a., 
• 
• 
• 
a„ 
a.y, 
a„ 
D 
^ 7’ 
■ 
• 
where D =77 -f- zd„ -f- fd t . Also, by similar processes, the above expression may be 
further transformed as follows : 
>- 1) 2 
where 
tv , 
7 , 
f , 
A, 
a„ 
V= r 
(« 
tv', 
Vi , 
w', 
to', 
B, 
a^j 
v' , 
7, 
v! , 
c, 
a„ 
l' , 
to', 
7, 
B, 
a,- 
D 
7’ 
A, 
B, 
c, 
B, 
• 
. 
a,, 
a., 
3,- D , 
• 
• 
A = 
1! 
a i 
tv', 
v' , 
7 , 
A, 
a« 
w', 
Vi 1 
7 , 
to', 
B, 
a* 
V 1 , 
v! , 
®>, 
tt! , 
c, 
a„ 
* , 
to', 
u' , 
, 
B, 
a. 
A, 
B, 
c, 
B, 
. 
. 
a„ 
a„ 
a c , 
a,, 
• 
. 
AY 
■ 2 - — y ) , 
n — 1 / ’ 
So that finally the expression □ 2 V=0, combined with V=0, C]V=0, is reduced to 
AV-fl + 2^ 
\ 11 — 1 
011 = 0 , 
