60 



On Gravity in relation to Centripetal Velocity. 



The following con- 

 struction (fig. 2) shows 

 p and its relation to 

 the elements of the 

 ellipse. PC=« = CA, 

 SC = e=CM=CF. 

 Draw PT cutting se- 

 micircle in Q! and Q. 

 Draw Q'R', QR, ± to 

 PA, ZTPC=p, 

 PR'=r 1} PR=/-, 



QR 



QR=\/V 2 -(«-r) 2 and pj= tan p; hence r 2 sec 2 p=2ar-b% 



and 



r=acos*p(l + \/l- -^-^}=acos 2 ^(H- ? ), (31) 

 L V a 2 cos 2 p J 



r^aco^p^-g). (82) 



This enables » . c^tanja -I L — > = — to be in- 

 tegrated as follows. Substituting in it the values of r and r, 

 given in (31), (32) and reducing, we obtain 



b 2 



because (1 — q 2 ) = 9 " Q : and since q= \ / \— _ . . 



v * ' a z cos 2 p' * V « 2 cos 2 j9 



d — = ff «^tanj9 .4t \/ jo — tan 2 j9= ^ 4 j-.d cos <j> sin <j> 



e 

 because maximum value of tanjt? is -r =Cm (fig. 2), and 



b*' 



tanjo=cos^>T # 

 In fig. 2 the angle TCft is <£. Hence for one complete revolution 



— J = — tt^t • 72 x area of semicircle radius unity 



c _ #e* 



(33) 



In the same way the integration of 



is effected, and thence is derived the changes in the other ele- 

 ments of the elliptic orbit as given above. 

 Inverness, September 5, 1866. 



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