[ 92 ] 



VI. 



SOME DIFFEEENTIAL PEOPERTIES OF THE OETHOGOJiTAL 

 TEAJECTOEIES OF A COXGEUEXCE OF CUEVES, WITH AJ^ 

 APPLICATIOX TO CUEL AXD DIVERGENCE OF ATECTOES. 



By EEGINALD A. P. EOGEES, F.T.G.D. 



Read Febbvauy 12. Published April 30, 1912. 



CONTENTS. 



A. Introductory. The family of curves (n) defined by Iclx + miy + ndz — 0. Nominal 



definitions. The complex of normal curves. Generalized geodesies, . .92 



B. Tn-o general principles of which Dupin's theorem and the allied theorems of Darboux 



and Joachimsthal are particular cases, . . ... 95 



C. Inversion and general conformal representation of a n-family. Effect on torsion of 



normal curves at a point, ........ 99 



D. Torsion and curvature of normal curves. The indicatrix of curvature. The indicatrix 



of torsion. Relations between curvature and torsion. Expression, by torsion and 

 curvature, of the quantities 



Idn dm\ Idl dn\ I dm dl\ , dl dm dn 



\Ty-l^.)^"'\dz-Tx)-^"\-d^-d,j) ^-^^ rfJ + ^ + rf:- 



where P + m- + n'' = \, . . . . . . . . 101 



E. Various theorems. DifiFerent geometrical expressions for the condition of integrahility, 107 



F. Second type of generalizations connected with Dupin's tbuorem, etc., . . Ill 



G. The indicatrix of form, . . . . .112 

 H. Geometrical expression for diveigence and curl of a vector by means of curvalure and 



torsion, etc., . . . . . . . . . .114 



A. Introductory. 



Let cj) («, 6, c . . .) be a true proposition involving certain entities or terms 

 a, b, c, etc. ; then it is natural to inquire whether there is not a more general 

 true proposition <f) {a, ft, y . . .), a being a special case of a, h of /3, etc. 

 Many other motives co-operate, but on the whole the desire to find more 

 comprehensive principles has been the main source of advance in pure 

 mathematics. 



The following investigation combines three generalisations. First, the 

 property of a line of ciunatiue is replaced by the notion of geodesic torsion, 

 which vanishes for a Hne of curvature. Secondly, in considering, for example, 

 Dupin's theorem, the variation and magnitude of the angle between two 



