140 Cambridge Philosophical Society, 



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 M be the symbol of any particular point of a right line whose 

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



M + rg, 

 r being arbitrary. 



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

 the symbol of the surface of a sphere is 



e being an arbitrary direction unit. 



3. If M be the symbol of any particular point of a plane, e and s' 

 the direction units of any two lines in the plane, the symbol of the 

 plane is 



M + rg+r'g', 

 r and r' heing arbitrary. 



4. If £ be the direction unit and r the numerical magnitude of the 

 perpendicular from the origin on a plane, the symbol of the plane is 



re + Dv-e. 

 V being an arbitrary line symbol, i. e. denoting in magnitude and 

 direction any arbitrary line. 



5. If M and u' be the symbols of two points, the symbol of the 

 right line drawn through them is 



u + m(u' — m), 

 m being arbitrary. 



6. If M be the symbol of any curve in space, the symbol of the 

 tangent at the point u is 



u -\- mdu, 

 m being arbitrary. 



7. The symbol of the osculating plane at the point u is 



u+mdu + m'd"u, 



m and m' being arbitrary. 



8. If s denotes the length of the arc of the curve, and e the direc- 

 tion unit of the tangent, then 



du 



