240 On the Motion of a Body about a Fixed Point. [Mar. 20, 



Hence, integrating, we get 



, T ^^-rw /vectorial area\ 

 ^ = G +aA \ GAD )• 



By subtracting the angle SGA, we get, if $ be the angle described by 

 GS in space, 



u T . . A /vectorial area\ 



APPEOT)IX. 



The following properties of the spherical ellipse will be useful in 

 connexion with what precedes. I have not been able to find them any- 

 where. 



1. Equation to the ellipse, 



, tan b , , 



tan y = tan y , 



J tana J ' 



cos a = cos y' . cos oc. 



Here y' is the eccentric ordinate and | an ^ is constant. 



tan ?/ 



2. The projection of the normal GN on the focal radius vector SG, 

 i. e. GL is constant and equal to semilatus rectum. 



If 1 = semilatus rectum, tan Z= ^ an - ; also ^ an ^ — constant. 



tan a sin GM 



3. If HAP cut GN at right angles, tan GN . tan GF = tan 2 b. 



4. The length GK cut off the focal radius vector by the conjugate 

 diameter is constant and equal to a. 



This follows from (2) and (3). 

 in 2 b 



5. If e 2 = l— . - , then G-M being an ordinate perpendicular to AN", 



sura 



tan AN = e 2 tan AM. 



6. Also S being a focus, 



tan (SG —a)=e tan AM. 



7. Polar equation to the ellipse, 



tanZ 1 e n £ A 



= 1— — — cos GSA. 



tan SG cos 2 b 



