262 
ON THE APPLICATION OF SURFACES TO EACH OTHER. 
[931 
where I recall that the values of w, w, w, X, and p" are aa + b0 + cy, a'a + b /3 + c y, 
aa' + 6/3' + cy', a 2 4- /3 2 4- y 2 , and aa' 4- /3/3' 4- yy', respectively. Observe that the first term 
in this expression for is 
We thus have 
dn 
dp ’ 
dm 7 E'ux — Fa> 
u = cos L — — 
dp VE 
dm . , E' 
V ~dp Sml ~VWE’ 
dn dl _ 2 \JE 
W dp dp \/A 
a , b, c 
a , ß , y 
ai, ft, yi 
+ (/¿'E - \F) - ^ (m'E - aF) \ , 
a, b, c 
CL, 0, y 
a, 0', y 
and we thence obtain at once the values of U, V, W; viz. these are 
Tr dM T Gen" - F*r" 
U = dq C0SL = VO ’ 
Tr dM . T G' 
T " dq SmL V^/G’ 
_dN dL 2 xJG 
dq dq 
+ 
Y 2 A' 
( 
a' , b' , 
c' 
) 
Uco'V - v"F) 
4, 0", 
y" 
+ O’ VG (jiG - \"F)--F- (v’G - w"F)\ 
{ 
0", 
n 
7-2 
. ) 
where a 2 ", ¡3", y" denote the derived functions of a", /3", y" in regard to q, viz. 
these are the third derived functions of x, y, z in regard to q. 
We have, moreover, 
dx 2 + dy 2 4- dz- = E dp 2 4- 2F dpdq + G dq 2 , = r 2 di? 4- 2rR cos 6 dtdT + R 2 dT 2 ; 
that is, 
r = ^E, R = \JG, cos 6 — 
F 
VEG’ 
and therefore also 
sin 6 
4EG - F 2 
4EG 
V 
VEG‘