DIFFEKEXTIAL INVAIMAATS OE SPACE. 
325 
Tlieii 
Now 
</'oio = 
^UUl = ^■‘•'^3 + 
C) = (A, B, C', I, G, 4*wi}' 
= + ^uiu ^2 + 
+ {4>umVi + 4’inoV-: “ ^ouGs)' 
+ {4^mCi + < 1 ^ 010^2 + 
on 
Now 
so that 
B, 
C, F, 
LB 
H; 
when 
further substitution 
we 
have 
(-), : 
^ + 
+ Gd) L-. 
dx 
dy 
dz 
d n 
(hi 
dn 
1 
hf. 
' % 
(hv 
+ ciy + fhd)’ ’ 
d(j> 
dn 
_ d(p 
dn 
= 
d\r ' 
B in + dn 
= 
(df. 
^ + dy- 
d- dv)'; 
0, 
_ /d(f)\ 
0 
1 
Id 
\dn/ 
! • 
'^001 
and therefore 
30. Let (In' denote an element of distance throngli the point in a direction normal 
to the snrface <j)' = constant ; and similarly let (.In" denote an element of distance 
through the })oint in a direction normal to the snrface (f)" = constant. Also let 
f/>' (//, r, u) = {X, y, :), (/A (a, r, iv) = G" (.r, y, :). 
Then, as 
21 .) _ (f)' 
+ 4 
uin 
'oiu 
^ 4*t)i)i 
21 ,)' _ (f} 
vre 1: 
c’^ioo 
7 C 1 / // 
UlU ^ ; ~r Y UUl ■^.1 ' 
^9uiu ^ruui 
ave 
.1J01 — {^lou^l d" <^ulu^2 d" Woolfs) i 4 *' ifo^l + 4 >' 010^2 d- </>' 001 fs) 
d- (<^100^1 d- (^010^2 d- <^ooi’?3) d- </>'orA2 d- ?'ouG:;) 
+ ('/'looCi d“ 4\mC2 d“ ^uui^s) lou^i d" 4 ^ 010^2 d“ ^ 001^3) 
= L- d- d- 
_ y .2(^l(f> cl(f)' f(lx dx' , dy dy' , dz dz 
dn dii! \dn dit dn dn' dn dn4 
= Id COS n", ~ 
dn dn 
