114 Proceeding!^ of the Hoi/al Irish Acudemii. 



at the point where it meets PQ. Thus to any direction VQ we have a fseudo- 

 conjugate direction PQ' ; but the pseudo-conjugate of PQ' is not PQ, unless 

 7=0, when we get the ordinary theorem of conjugate directions. If the 

 coordinates of Q are .r, y, 0, a- and y being small, and a, /3', the direction- 

 cosines of the pseudo-conjugate of PQ, the conditions give 



aXa^x + cuy) + ^'{h^x + h^y) = 0. 



Comparing this with the differential equation for the indicatrix, the theorem 

 is proved. 



(6) The directions of zero normal torsion are those which are perpendicular 

 to their pseudo-conjugates. 



(c) The inflexional directions are those of the two right lines | = 0, i? = 0, 

 which are included in the indicatrix, and are asymptotes of all its curves. 



Curves on a Surface. 



The points of contact of a surface U = Q with a Il-family may be defined 

 as the points where the surface touches the tangent plane of the family, and 

 are the intersections of the surface with the curve 



IdU -i_dV' 1 dU 



I dx rii dy n dz 



From the preceding it would appear that all the curves of the singly 

 infinite system in which the surface cuts the Il-family collect spirally round 

 the ' spiral ' or ' elliptic ' points of contact, and that one curve passes through 

 each hyperbolic or non-spiral point, having a double point thereat. 



H. Geometrical Expression for Divergence and Curl of a Vector 



BY MEANS OF TORSION AND CURVATURE. 



A vector is defined by a direction {l,rn,,n) and a magnitude B, i.e. by 



the quantities X, Y, Z, where X=IE, Y = mP, Z = nE. If X, Y, Z are 



functions of tliree varialiles, we can associate a vector with each point in space. 



The Curl of the \'ector is the vector the magnitudes of whose components 



dZ dY ,1-1^. p , ■ , .1 



are -j j^, etc., and the Divergence of the vector is the magnitude 



f?X dY_ dZ* 

 dx dy dz 



* a. Aleni&iaudaiu on Notatioti in Whittaker's Historij of the Theories of Ether mid Elcctricit;/. 



