60 Mr. L. V. Meadowcroft on the Curvature and 
vertical plane perpendicular to the meridian plane at P, 
yjr being given by tani|r = -^=-— . 
The tangent line at P lies in the tangent plane at P and 
makes an angle a with the vertical. It easily follows from 
this that cos % = tan ty cot a, x being the angle which the 
vertical plane containing the tangent line makes with the 
meridian plane through P. 
At Q x w iU have the value x -f dy^ where 
— sin x d% = se ° 2 ^ co ^ a c ty- 
•*• dco = d(f>+{x + dx)-X 
= # + ^% 
sec 2 -dr cot a 7 , 
= d6 * d4r. 
r sin x 
Now 
tan ^ = /fe 
. • . sec 2 >lr dtlr — — 'Arm dr. 
J\ r ) 
cot a f r (r) , 
dco = a<i + — — 'yrr-\o dr 
r*/tan a «./'fr) 8 -l _cota__ __ /V)~U 
" ± L r v^l-tan 8 yfr cot 2 a f(r)*J 
Now 
1 
p 
dco 
= sina^— 
ds 
dco 
= sin a -r- 
dr 
dr 
'dz 
is 
i 
= ±sin a. .- 
M 2 i 
*/(r) 3 - 
-/'('')+'':/"('•) 
1 
r/ / (r)4/tan s »./ r (r) , -l 
Y'M 
cos a. 
^ * ~~ tan 2 a ./'(r) 3 — /'(>) + r/"(r) ' sin a cos a* 
Again, 
1 <Za> 
- = cos a-; — 
<r as 
rf(7Q 2 \/tan 2 a./ v (r) 2 -l 1 
~ ± tan 2 a . /' (r) 3 -/'(r) + r/"(r) * cos 2 u 
