368 
PROFESSOE A. E. FORSYTH ON THE DIEFEEENTIAL INVARIANTS 
1 cos uT . 
r = = A, 
P P 
1 sin CT 
P P 
I c/trr ITT 
T = <fe - ’ 
I = - W, 
r 
i (y) = (P’ Q- P- 2/')’. 
f/fii p’) ~ y’ y')*; 
111 tli0 Icist t^^o ccjuntioiis X and y ar© used in jilac© of clxldt and dyjClt^ 
tliey are equal respectively. The relation 
to 
at once gives 
and we also have 
= ah - - K 
J 
Pi 
I 
r r' - p'-^ + p-^ 
' = ' + 
f-i ' /o ’ 
P P P 
so that 7, p, and 11 are expressible in terms of p, p", t and of their derivatives. 
’)( 
' 1 
- 1 ') . 
p' 1 ' 
' P' 
P2' ’ 
. d" 
dp> 
ds 
/ dp 
X 
33. As regards -f ( ■ 
and therefore 
The Values of 
■ ds ds dn dn 
x') and (= ;?/'), we have 
4>\iv' + 4>i)\y' = 0 , 
+ 2F.rV' + Gy'-^ = 1 ; 
4*0 
r = V — — 
^ u — 
v/ a'n 
which 
Niext, differentiating along a direction in the surface that is perpendicular to the 
