236 



Mr. E. J. Routh on the Motion 



[Mar. 20, 



If, therefore, 1STL be a perpendicular arc drawn from N on SG, we 

 have 



tan GL = tan GN cos SGN. 

 Hence the resolved velocity of G 



r 



r (G 2 -BT)(AT-~G 2 ) 



G 2 -CT, 7 

 -CG- tanZ 



AB 



Hence throughout the motion the extremity G of the invariable axis moves 

 along its ellipse in such a manner that its velocity resolved perpendicular to 

 the focal radius vector SG is constant. 



8. Let us next find the velocity of G resolved along the focal radius 

 vector SG. Let SG=p, GSA=fl; then the polar equation to a sphe- 

 rical ellipse may be proved to be 



sin « cog a (W) = cos „ b _ e cos _ 

 tanp 



Differentiating this, we get 



(1— e 2 ) sin a cos a dp . _ dd 



± L— J- =—e sm d — . 



sin p at dt 



dd 



But the value of sin p~ , being the velocity of G perpendicular to the 



cct 



radius vector S G, has just been found. Also if GM be drawn as an 

 ordinate perpendicular to the major axis, we have sin p sin 0=sin GM. 

 Hence, substituting from art. 3, we get 



dp G / (A-CXB-C) ^ GM< 

 dt C V AB 



Hence the velocity of G resolved along either focal radius vector bears to the 

 sine of the ordinate of G a ratio which is constant throughout the motion. 

 This constant, when divided by the resultant angular momentum, is inde- 

 pendent of the initial conditions. 



9. We may also refer the motion of G to the centre of its ellipse. 



The velocity of G along its ellipse being — — - — tanGN, the re- 



solved velocity perpendicular to AG is 



= a2 ~ CT tan GIST . cos AGN. 

 CG 



But if A~F be an arc drawn from A perpendicular to GN, we have, 

 by Napier's rules, 



cos AGN =tan GF . cot AG ; 

 and in any spherical ellipse we have 



tanGN.tanGE=tan 2 5. 



