136 Messrs. Galbraith and Haugliton on the Apsidal Motion 



nitude and dii*ection of whose axes ai-c continually varying. In 

 order to see how R produces these effects, suppose it resolved 

 into two components, 11 cos <^ and R sin <^ : the fonner, acting 

 along the tangent, either accelerates or retards the elliptic velocity; 

 whilst the latter, acting in the direction of the nomial, increases 

 or diminishes the angle <^ which the tangent makes with the 

 radius vector. These variations of <\> and v arc the immediate 

 effects of the disturbing force. The momentary variations of the 

 elements of the orbit depend on and may therefore be expressed 

 in teniis of them, so that by the application of the rules of the 

 integral calculus we can calculate the total variation of these 

 elements produced in a given time. '^' ""'" 



To fix our ideas, suppose that P is in the fifst quadrant moving 

 from the apse A towards B, the effect of the tangential compo- 

 nent is to diminish the velocity in the ellipse, and that of the 

 normal component to increase the angle which the tangent makes 

 with the radius vector ; these variations are expressed as follows : 



8.= -R cos </,<;< S,^=?^lEi*. . . (3.) 



From these and equations (1.), we obtain 



e, R cos <f) sin^ r«?a) 

 tv= J- ^ 



5., R sin^ <f> sin^ r</G) 



«*= — -^^ — 



It may be easily shown that 



tan^ r sin 2ft) = tan^ m sin 2ft)' (4.) 



Differentiate this, considering r constant, and eliminating W by 

 means of the relation 



cosrtan<^= tan (ft) + G)'), (5.) 



we obtain after some reduction 



(a^— ^^)Sft)=sin2ft)'tanmStanm j 



„ ,, . cosM<w + «') 5^^ I ' * ^^'^ 

 + cos2a)'tan27w ^ . - cos r6<j> 



It may be easily proved from equations (1.) that 



tan^m= — cos*^ (7.) 



By means of this and equations (3.), we may eliminate B tan m and 

 B^ from (6.), and obtain, finally, 



/ O /VJNC* R. 7 sin (ft)' — ft)) g /Q X 



(a^—6^)B^= —tsmrdco ... ■ ; cosV, . . (8.) 



^ '^ ^ (/ sm^a^-t-ft)) 



1 



(3.) 



