Cambridge Philosophical Society, 141 



9. _ or — d{ — \ represents a line equal to the reciprocal of the 

 ds ds \ dsj 



radius of curvature drawn from the point u towards the centre of 

 curvature, i. e. it represents what may be called the index of curva- 

 ture in magnitude and direction. " 



Hence, since « = xa + y/3 + zy, the numerical magnitude o^ y ^ ( ;r- ) 



which is the general expression for the reciprocal of the radius of 

 curvature. 



10. The symbol of the normal which lies in the osculating plane is 



u+md 



/du\ 

 [Ts)' 



m being arbitrary. 



1 1 . The symbol of any normal at the point u, i. e. the symbol of 

 the normal plane, is 



M + Dy.rfw, 



V being an arbitrary line symbol. 



12. The symbol of the normal perpendicular to the osculating 

 plane is 



u-\-rnDd"~u.du, 

 m being arbitrary. 



13. If M be the symbol of a surface, involving therefore two vari- 

 able parameters, A and /* suppose, then the symbol of the normal at 

 the point u is 



TV du du 

 u + mD -— . — -, 

 d\ dji 



m being arbitrary. 



14. The symbol of the tangent plane at the point u is 



J , du , du 



u+mdu, or u-\-m 1- n — , 



d\ d^h 



m and n being arbitrary. 



15. The symbol of the plane which contains the three points 

 u tt' u" is 



u + m{u'—u) + n{u"—u). 



16. If M be the symbol of a right line, the symbol of the plane 

 containing it and the point u' is 



u+m(u'—u). 



The following are examples of the proposed mechanical system in 

 addition to those given in the paper already quoted. 



1 . If r be the radius vector of a planet, and a /3 y be chosen so 

 that a is the direction unit of the radius vector, and y perpendicular 

 to the plane of the orbit, it may be shown immediately by the sym- 



