304 The Rev, J. A. Coombe on the Motion of the Apse-Line 



depends upon the principle of the Variations of Elements so beau- 

 tifully applied by Lagrange to the planetary perturbations. 



Let P be the ball of the pen- 

 dulum ; K the point of suspen- 

 sion ; PN perpendicular to the 

 table; PL parallel to AN. 



LetKP==/,KL = r,AN=r, 

 T the tension of the string KP. 

 Then the force upon P resolved 

 in the projection AN 



.T.J{,|.M!}, 



and 



(vel)«=2^-^ + C and z=: V^/^-r^; 



/, force to the centre A 



omitting higher powers of r than the third ; or the accelerating 

 force to centre 



The second term may be looked upon as expressing a small 

 disturbing force upon the first, which alone would cause the pro- 

 jection of P to move in an elliptic orbit about centre A. Hence 

 by Lagrange's principle we may consider the motion as taking 

 place in an ellipse with variable elements ; so that if the disturbing 

 force were at any moment to cease, the body would go on de- 

 scribing an ellipse about A as centre with elements corresponding 

 to their value at that instant. 



Let the equation to the instantaneous ellipse be 



cos^((9-tg) , 

 Then we have, by the theory, 



sm- 



(l9-'sr) 



But 



and 



1^ 



7^ 



0=^ 

 da 



COS' 



da 

 dt 



+ 



b^ 



dr db . dr 



+ -,- • -TT 



db dt dvr 



dm 

 dt' 



(1.) 



^-w 



a« 



dr 

 db 



sin^^- 



b^ 



I dr . -3 . /I 1\ 



1 ' «fcr \ : r- ^ V V, / . < \b^ a^/ 



