502 Mr. O'BRIEN'S CONTRIBUTIONS TOWARDS A SYSTEM 



(du\ 

 of the elementary line joining the ends of these two direction units will be di—j . Now this 



elementary line is evidently parallel to the normal drawn to the centre of curvature, and numerically 

 equal to the angle of contingence (as the angle made by two consecutive tangents is commonly 

 called). 



24. Let V be the numerical magnitude of rff — J , e its direction unit, and p the radius of 



curvature ; then, according to a well-known theorem, 



ds 



Hence, we may immediately deduce the well-known expressions for p. 

 We have u = xa + y^ + xy. 



Hence 



ds 



which is the well-known expression for p, the independent variable being arbitrary. 



25. If TO be any number, it is clear that m,dl — \ represents in magnitude and direction 



any line drawn from the point of contact through the centre of curvature. Hence, the formula of 

 that normal which lies in the osculating plane is 



u + md [^) , 



m being the variable parameter. 



26. The symbol of the centre of curvature is evidently, 



ds 

 ti + — 6, 



(du\ 

 £ being the direction unit, and v the numerical value of d ("p I • 



27. If S denote any arbitrary variation (as in the Calculus of Variations), then S i—j denotes 



any small line at right angles to the direction unit — , i.e. to the tangent. Hence, the formula 

 of any normal at the point u is 



It is obvious that this is also the formula of the normal plane, for it is the symbol of any point 

 in the normal plane. 



