ON THE CAPTURE OF COMETS BT PLANETS. 515 



comet was at T when tlie planet was at O, aud the planet describes 3 A. 

 while the comet describes YA, leaving BC as the distance yet to be 

 described, or h. But the angle CBD is 0, so that we have 



_p2 = 0D2 = 0C2 + CD2 = d-^ + /i2 sin2 9 . . (6) 



10. To find a. — The angle a is the acute angle between the asymptote 

 and the trans vei'se axis of the hyperbola, and hence from the nature of 

 the hyperbola tan a = B/A. By known formulas we have, if the planet 

 is at its mean distance. 



Therefore 



Hence from (6) 



tan a = 





t/'o'' mr mr 



„ 2 = X' °^ -^ = T.2 



A~A mr 



(7) 



11. To find <f>. — The orbit of the comet i-elative to Jupiter lies in the 

 plane YOB. Let i be the inclination of the plane YOB to YOX, 

 measured positive from x positive to z positive ; let I be the longitude of 

 the direction YO, measured in the plane YOX from OY, that is, the 

 angle made by YC with OY produced; let A be the longitude of the 

 direction YB measured in the plane YOB from OY, that is, the angle 

 made by YB with OY produced. Imagine now a sphere described 

 about Y as a centre that shall cut the three planes XOY, BOY, and BCY 

 in three sides of a right-angled spherical triangle. The hypotenuse of 

 this triangle is X, the base I, the perpendicular ^tt — 6, and the angle 

 opposite to the perpendicular is i ; hence we have 



cos A = cos Z sin ^ . . . . (8) 



cos ^ = sin t sin A . . . . (9) 



cot i = sin I tan 6 .... (10) 



Also from the triangles OCY and BCY 



OO (I 



tan I = tan OYC =^ — ^-,= — ,- - ,, • . . (11) 

 X o n la u !) 



The angle <f> is by definition the angle between the direction OE and a 

 line in the plane YOB that makes with YB an angle a. Hence we have 

 readily 



cos cf> = sin i sin (^ ± °-) • • • • (1-) 



These equations enable us to compute <j) in terms of d, h, aud m ■ for in 

 succession 6 may be computed by (3), I by (11), A by (8), i by (10), aud 

 <f> by (12). 



12. These values of s, p, a, and cp give by equation (2) the value of @. 

 The suppositions that the planet is at its mean distance, and that , is a 

 parabola, are involved in that equation, but they are not necessary o the 



L L 2 



