39 



In this view of the case, the author has entitled his paper Contri- 

 butions towards a System of Symbolical Geometry and Mechanics. 

 The proposed geometrical system consists, first, in representing 

 curves and surfaces, not by equations, as in the Cartesian method, 

 but by single symbols ; and secondly, in using the differential notation 

 proposed in a former paper* to denote perpendicularity , and to ex- 

 press various equations and conditions. The proposed mechanical 

 system is analogous in many respects- Examples of it have already 

 been given in the paper just quoted. 



The author uses the term direction unit to denote a line of a unity 

 of length drawn in any particular direction ; and he employs the 

 symbols a (3 y to denote any three direction units at right angles to 

 each other. 



He defines the position of any point P in space by the symbol re- 

 presenting the line OP (O being the origin) in magnitude and direc- 

 tion . If x y z be the numerical values of the coordinates of P, and 

 a /3 y the direction units of the coordinate axes, the expression 



cca+yfi+zy 



represents the line OP in magnitude and direction, and therefore 

 defines the position of P. This expression he calls the symbol of the 

 point P. 



If r be the numerical magnitude, and s the direction unit of OP, 

 we have 



rs=xa-\-yl3 + zy : 



r s is therefore another form for the symbol of the point P. 



The following is the method by which the author represents curves 

 and surfaces. 



If the symbol of a point involves an arbitrary quantity, or, as it is 

 called, a variable parameter, the position of the point becomes inde- 

 terminate, but so far restricted that it will be always found on some 

 line or curve. Hence the symbol of a point becomes the symbol of 

 a line or curve when it involves a variable parameter. 



In like manner, when the symbol of a point involves two variable 

 parameters, it becomes the symbol of a surface. 



The parameters here spoken of are supposed to be numerical 

 quantities. An arbitrary direction unit is clearly equivalent to two 

 such parameters ; and therefore, when the symbol of a point involves 

 an arbitrary direction unit, it becomes the symbol of a surface. 



The following are examples of this method : — 



1 . If u be the symbol of any particular point of a right line whose 

 direction unit is e, then the symbol of that right line is 



u + rs, 

 r being arbitrary. 



2. If u be the symbol of the centre of a sphere, and r its radius, 

 the symbol of the surface of a sphere is 



u -f rs, 

 e being an arbitrary direction unit. 



* Read Nov. 23, 1846. 



