931] 
ON THE APPLICATION OF SURFACES TO EACH OTHER. 
257 
and similarly the other two parts are 
(- Ca + Ay) 2 and (- A/3 + Bg) 2 ; 
the equation is thus verified. 
wj • i •? dni/ dn dm , dn 
VV e require Godazzi s ^ and , or say ^ and ^; these are to be obtained 
from the equations 
d_ jx _ Ea. — aw dm d by — c/3 _ Eg — aw dn 
dp \/E VEA dp’ dp \/A VEA dp’ 
and the values obtained should satisfy 
d Eg — aw_ a dm by — cfi dn 
I find 
dp V EA \/E dp aJA dp ' 
a , b , c 
dm _*J A dn _ aJE 
dp É ’ dp A 
a , ¡3 , y 
a i> fti> 7i 
where a 1} /3 1; y l are the derivatives of g, ¡3, y in regard to p; and the equation 
to be verified thus is 
a , b, c 
d Eg — aw _ a*/A (by — cA)rJE 
dp \ f EA E aJE A aJA 
« , /3 , y 
a i > fii, 7x 
dm 
First, for : the derivatives of a, E are a and 2 (aa + b/3 + cy), = 2« ; we thus 
have 
Avhich is 
viz. we have 
d a 
cico Eg — am 
dp \JE ’ \/E E \/E E ~aJE ’ 
_ Ea — aw dm 
V EA dp 
dm _ aJA 
dp E 
dn 
Next, for using a subscript Q to denote derivation in regard to p, we have 
A = EX — co 2 , and thence 
= EiX -f- EXi — 2wwj 
= 2wX + E. 2 (gg 1 + ¡3(3-l + yy x ) — 2w (X + aaj + bfr + cy x ), 
= 2 [E (gg 1 + /3& + yy^ — w (aa x + bfi x + 071)}, 
= 2 (ajX + /3 2 F + 7jZ), 
c. XIII. 
33