1890.J and the Curvatures of their Trajectories. 39 



These are the equations of the path of P, correct as far as the 

 second order, and referred to the parameter t. 



The determination of the radius of curvature E at the origin 

 may now be made by the usual methods, bearing in mind that 

 the angle between the axes is &>, the angle between BB' and GO'. 

 It is sufficient to give the result 



R ( by 2 + cy+2bcppcos^ 



a sin a (be — be) pp 



for its interpretation suggests a purely geometrical method of 

 arriving at it, as follows. 



The velocity V of P' is the same as that of N, and is therefore 



the resultant of velocities — b and - c parallel to BB' and BA : 



a a L ' 



thus 



V 2 = - 2 (b 2 p 2 + c 2 p 2 + 2bcpp' coso) (11). 



CI 



In the same way, the acceleration / of P' is the resultant of 

 accelerations — b and - c parallel to BB' and BA ; its value may 



Qj Co 



therefore be written down. 



Also, if 6 denote the angle between V and f, we have Vf sin 9 

 equal to the area of the parallelogram contained by the vectors 

 representing Fand/; therefore 



Vfwa.6 = (£ b?-6-£c P -b)sma> 



\a a a a J 



= £§ (bc-bc)s'mo3 (12). 



Now by Huygens' fundamental formula of centripetal acce- 

 leration in a curve, 



/sm0=^- (13), 



therefore we arrive at the formula (10), which may also be 

 written 



a 2 V 3 



R= ~ . —, (14). 



(be - be) sin co PP 



We may exhibit the result in a geometrical form. 



