40 



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

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

 plane is 



u + rs + r's', 

 r and r' being arbitrary. 



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

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



rs-\-Dv.£, 



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

 direction any arbitrary line. 



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

 right line drawn through them is 



u + ?n(u' — u), 

 m being arbitrary. 



6. If u 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 4- mdu + m'd^u, 

 m and m' being arbitrary. 



8. If 5 denotes the length of the arc of the curve, and e the direc- 

 tion unit of the tangent, then 



du 



ds 



9. — or — dl — J represents a line equal to the reciprocal of the 

 ds ds \ ds/ 



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 u = xa. + y(3 4- zy, the numerical magnitude of — dl — J 



ds \ds/ 



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\ 

 Is)' 



m being arbitrary. 



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

 the normal plane, is 



u + Dv.du, 



v being an arbitrary line symbol. 



