478 
MR. A. B. BASSET ON THE EXTENSION AND FLEXURE OF 
We shall now find an expression for the change of curvature along any great circle 
which makes an angle y with a meridian. 
In the figure on p. 463 join OD, and let the angle OED = y, and the angle 
DOA = y ; then by (45) the change of curvature along OD is 
1 
dho 
2 \ 7 
\ dx 
If iv w be the normal displacement at D, it follows by equating the two values 
of that 
From the spherical triangle ODP we have 
— cos y — 
cos PD — cos 6 cos Sy 
sin Q sin Sy 
whence 
Agi 
am 
whence 
sin 6 SO = cos y sin ^ Sy + 2 ^ “ 2 ^ 
= cos y sin ^ Sy + ^ cos 0 sin^ y Sy^, 
sin S(f) sin 7 
sin 3y sin (6 + S0) ’ 
Substituting these values of SO, Scf) in (46) and equating coefficients of Sy^, we 
obtain 
dho siTOydho , sin 27 dho . „ dho sin 27 cos 0 div , . „ , ^dw , . . 
+ ~::^:S:77+C0S'r^^,-- + Sm=>yCOt (47). 
d-x sin^ 0 d(f)^ sin 9 dOdcf) 
sin- 6 d(f) 
dd 
Whence it follows, that if p^, p, 2 , are the principal radii of curvature along and 
perpendicular to a meridian 
1 
1 
1 
(dho . ' 
— - 
- — 
1 + W 
p\ 
a 
a? 
1 
1 
1 
( 1 dho 
Pi 
a 
Vsild 6 d(f)^ 
+ cot 0 
dw 
de 
4- w 
(48). 
