Manchester Memoirs, Vol. hi. (19 12), No. 4. 



IV. To FIND THE Curvature and Torsion of 

 ANY Curve drawn on a Sphere. 



As an example of the use of the above formula; they 

 may be emplo}'ed to determine the curvature and torsion 

 of any curve on a sphere, since every curve on a sphere is 

 a line of curvature. In this case the first formula breaks 

 down, as we should expect, since we must express in 

 some way the orientation of the curve. The direct 

 calculation of the curvature, however, can be made very 

 briefly by means of a simple device. Let (x, j\ .:) and 

 {x-\-dx, y + dy, rj-\-da) be the coordinates of two neigh- 

 bouring points of a curve on a sphere of unit radius and 

 let (y, (/»), {B-\-dd, (f)-\-d(p) be their angular coordinates. 

 Let the curve cut the meridian at an angle w. 



Then 



cosw = — , suioj = sin6— L 

 ds ds 



.... (v). 



Now x = s'\nQ cos (p, ;;' = sin 0sin ^, .:. = cosy. Hence if 



(/, JH, H) be the direction-cosines of the tangent line at 



[Q, (f)) we have 



do . . dih 

 /= cosycos^-^ -smt/sm^y; = cosycos(^cos w-sm^sinw , 



dB . d(h 



tti = cosy sin0-p + sm6cos0— = cosflsm^cosw -f- cosi/) sin w, 



sinf) 



d^ 



ds' 



- sin cos w, 



from 

 (V). 



dl ■ f.dd) d . ^ ^ 



.— = - sm H--COS 0COSa) + cos(i^(cos6cosw) 

 ds ds ds 



ds 



dm ii(h , • '^ , r. \ 



-- = cosf/> -^ cos0cosw-f sm0— (cos cos w) 

 ds ds ds 



dd) . . dw 



— cos 0-^ sin w - sin ^ cos w— > 



ds 



dd) . dii) 



sind)— sin w -I- cos(i)Cos(wv-' 

 ^ds ^ ds 



dn ^dB ■ .. ■ dio 



-- = - cos B-— cos w + sin sin ta -j-- 

 ds ds ds 



