41 



12. The symbol of the normal perpendicular to the osculating 

 plane is 



u-\-mDd q u.du, 

 m being arbitrary. 



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

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

 the point u is 



-^.du du 



u -+- mD — , 



d\ d/x 



m being arbitrary. 



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



7 du , du 



u + mau, or u-\-m \-n— — , 



d\ dy, 



m and n being arbitrary. 



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

 u u' u" is 



u + m(u' — u) + n(u" — u). 



16. If u 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- 

 bolical method, that the symbol of the force acting on the planet is 



A , 1 d(r q w) n . , 



l )a.+ 7 ' B + rwcoy, 



J r dt 



where cu is the angular velocity of the planet, and cu' that of the 

 plane of the orbit about the radius vector. The expressions for the 

 three component forces along r, perpendicular to r, and perpendicular 

 to the plane of the orbit, are the coefficients of a j3 y in this expres- 

 sion. 



2. The equation of motion of the planet, when the force is the 

 attraction of a fixed centre varying as the inverse square of the di- 

 stance, is 



d*u jua 



~dF~ ~ !*' 



It is curious that this equation is immediately integrable, the in- 

 tegral being the two equations 



h 

 r 



The latter equation is the symbolical equation of a conic section, 



£ +eAj3.e. 

 r h 



