SPECIAL CASE. 



89 



V. SPECIAL CASE i=0 i,=0 a^=0 e=0 S =0* 



This means that two bodies undisturbed by any exterior force revolve 

 in circles, that the radius, mass, and angular velocity of rotation of one of 

 them are so small that its rotational momentum and energy may be 

 neglected, and that the axis of rotation of the other is perpendicular to the 

 plane of their orbit. In this case equations (9) and (10) become, writing D 

 in place of D^, 



(23) 



M = PhJ^^ 



E 



71 



1 



W, 



pi ' D' 



(24) 



We may choose the direction of revolution of the bodies as the positive 

 direction. Then only a positive P can have a meaning in the problem, 

 since a revolution in one direction can not be reversed without a collision 

 of the bodies. D is positive or negative according as the rotation is in the 

 same direction as the revolution or the opposite. Under the hypotheses 

 adopted M is rigorously constant. When D=+oo then P=M^; when 



D = -^then P = 0: when D = lim(0-\-£) then P=-oo; when D = lim(0—e) 



then P = + 00 ; when D = — cc then P = M\ Consequently the curve 

 defined by (23) is as given in fig. 9. The part of the figure to the right of 

 the P-axis belongs to the case where the rotation of m-i and revolution of 

 mj are in the same direction, and the part to the left where they are in 

 opposite directions. 



,P 



Fia. 0. 



The slope of the curve, or the ratio of the rate of change of the period 

 of revolution to that of rotation, is found from (23) to be 



dP / dD Sm^Pi 



dt 



dt 



D' 



(25) 



♦ For a similar treatment of this problem see No. 5 of Darwin's papers. 



