102 Proceedings of the Royal Irish Academy. 



with regard to r,y,z, vanish owing to the relation l^ + m' + n- = 1. Also 



at the origin 



dl dl , dm 



«, = -^, a. = — , bi = -5-, etc. 



ax dl/ rfa; 



The Indimtrix of CurvatKre. 



The n-iiidicatrix of cm-vature at a point F is simply the Dupin indi- 

 catrix of the surface {S), generated hy the normal curves thi'ough P ; but 

 since this surface varies from point to point (even if we move from P along 

 the surface k> associated with it), the Il-indicatrix of curvature is a more 

 general conception than the Dupin indicatrLx of a one-parameter family of 

 surfaces. 



Let - Vie the curvatui-e of a Il-curve whose tangent-line normal and bi- 

 P 

 normal are a, /3, 7; !,m,n; X,tJi,v, then, by the Frenet-Serret formulae,* 



dl fa X\ ^ 

 ■ ds^-K'p-'r} '''■' 



... 1 / dl ^ dm dn 



and therefore -=-a-5- + p-r + 7^- 



p \ as "^ ff-s as 



Using the special axes a = cos 6, /3 = sin 0, 7 = 0, where 9 is the angle 

 between the tangent line and the axis of x. Therefore 



1 dl dm . 



- - = ^- cos b + - - sin b. 

 p (Is as 



Xow ' 



dl dl , dl dl ,. . „ 



-rr = a-r- + P -r— + J ^T = ^'l COS ff + dj Sm 0, 



ds ax d,y d,z 



and -T- similarly = 6, cos + &2 sin i). Therefore 

 ds 



= «, cos- ti + («2 + ii) sin cos 6 + hz sin^ 9. 



P 



The conic a^x' + [cu + bijxi/ + h^' = constant may bfe described as the 



indicatrix of curvature at the poiut. It may be supposed to lie in the 



tangent plane at the poiut ; and its properties as regards cui-vature are similar 



to those of the Dupin indicatrix. Eeferred to its axes it may be written 



a-' if 



— + — = const., 



Pi P2 



* Salmon, op. Ht., Art. 368 (o). 



