514 



REPORT 1891. 



8. To find s. — In fig. 1 let A and E represent the two points A and E 

 as defined above (Art. 4), and the line AE represent d. Let AY be the 

 tangent to (![,i at A, and EO the tangent to gl at E. It is an admissible 

 supposition that the planet is describing the straight line OE, and that 

 the comet in its unperturbed orbit is describing the straight line TA. 

 At some certain moment the line joining the planet and the unperturbed 

 comet must evidently be perpendicular to OE. Let OY be the line 

 joining the bodies at that moment, so that the planet is at O when the 

 comet is at Y, and EOY is a riyht angle. Instead, however, of supposing 

 the planet to move from towards E we may apply an equal, opposite 

 motion to the comet, and consider the planet to 

 remain at rest at O. Draw AC parallel to EO and 

 make AB equal to the distance described by the 

 planet during the time that the comet is moving 

 from Y to A. Join YB. Then since YA and BA 

 jepresent in direction and magnitude the motions of 

 the two bodies in a given interval, the third side YB 

 of the triangle represents in magnitude and direc- 

 tion the motion of the comet relative to the planet. 

 The angle YAB is the angle o>, and the three sides 

 of the triangle YA, YB, and BA are proportional 

 to V, z'o, and v^. Let the angle YBC be 6; then from 

 the triangle YAB we have 



Fig. 1. 



2 ^ ,. 2 — o,, 



and 



:i';v cos <j> + 7"', 

 V : V/ -.Vq: : sin 6 : sin {$ — (1)) : sin 



(3) 



Since V and v, can be computed from the given elements of the orbits of 

 the planet and comet, we may readily compute from w the value of s, 

 or Vajv,. But if the planet is at its mean distance from the sun, and the 

 comet's orbit is "parabolic, v"^ = 2y^^, and we have 



62 = 3 - 2^/ 2 cos w . . . . (4) 



Also from the triangle 



2z;/ = -y„2 +2r^v,cose+ v,\ 



2s cos 61 = 1 - 6^ 



(5) 



9. To find p. — The planet being regarded at rest at O and the relative 

 un|itrturbed motion of the comet being along YB, this line may within 

 admissible limits of error be treated as one asymptote of the relative 

 orbit C. The perpendicular from O upon YB will then be by definition 

 (Art. 4) the line p. Draw OX from perpendicular to OY and OE, 

 and let these three lines be coordinate axes. Let the line AB meet the 

 plane XOY in C. Join OC, let fall OD perpendicular to YB, and join 

 CD. Since EA is perpendicular to AY, and also to EO, and so to its 

 parallel line AC, therefore it is perpendicular to the plane YAC. Hence 

 OC parallel to EA is perpendicular to the plane, and so perpendicular 

 to CD. Again CDY is a right angle ; for OD^ + DY^ = OY^ = OC^ 

 + CY2, and OD^ = OC^ + DC^. Hence DC^ + DY^ = CYS and conse- 

 quently CDY is a right angle. 



The quantity h is the line BC ; for h is the distance which the planet, 

 when the comet is at A, has yet to pass over before reaching E. But the 



