14 Proceedings of the Royal Irish Academy. 



10. The equations of motion, as found in § 5, may also be expressed in the 

 polar coordinates (p, <p), and they become 



dr-p /d<pY d^> dV 



d ( „<fy\ T dp dV 



m^m-^ 



a form directly adapted to the above transformation. The transformed 

 equations are equally well adapted to Sir George Darwin's method of 

 mechanical integration. For, if J? be the radius of curvature in the 

 (£, »j) plane, 



dV »,' cV, 2nJ 



1 



»"r - rv 



E 



(T + W 





1 1 V 



P _ 2hJ" 



where v- = 2 V and P is the component of force normal to the trajectory. 

 Hence, if i£ is the inclination of the normal to the axis of £, and a is the 

 arc of the orbit from a chosen point, 



P = cos \P — sin $ -zj , 



and -,/, = ^ + -^ 



£ = £ + / cos -<f, da 

 jj = »)o + / sin ^ rf«r. 

 A final quadrature will give 



t = 



Jtfr = 



■Ld,. 



V 



