512 REPORT 1891. 



this sphere in a conic section of which the sun is the focus, and as 

 moving when within the sphere in a conic section (an hyperbola) of 

 which the planet is the focus. In other words, only perturbations of the 

 first order of magnitude are taken account of. A comet is treated 

 throughout this paper as a small indivisible body whose mass may be 

 neglected. 



4. Notation. — The symbols used in (1) and also other symbols which 

 I shall have occasion to use may be thus defined : — 



Let C^ be the orbit of the comet about the sun before the comet 



comes under the appreciable action of the planet ; 

 C the orbit of the comet about the sun after perturbation by the 



planet ; 

 C the hyperbolic orbit of the comet relative to Jupiter when 



near the planet ; 

 3> the elliptic orbit of Jupiter about the sun ; 

 A the point on <£, which is nearest to 3; 

 E the point on B which is nearest to <£■, ; 

 d the length of the straight line EA, being the perpendicular 



distance between the orbits at their nearest approach ; 

 (1) the angle between the tangent of C^ at A and the tangent to 



Sat E; 

 h the distance which the planet has yet to pass over to reach E 



when the comet is at A {h may be negative) ; 

 m the mass of the planet ; sun's mass = unity ; 

 a the unit of distance, in general the mean distance of the earth 



from the sun ; 

 / the sun's attractive force at the unit of distance ; 

 Vj the planet's velocity in its orbit at E ; 

 r^ the comet's velocity in its orbit C when the comet enters the 



sphere of Jupiter's perceptible influence ; 

 the comet's velocity at A relative to the sun ; 



@i the semi-axis major of (C, (negative if C/ is an hyperbola) ; 

 @ the semi-axis major of (j (negative if d is an hyperbola) ; 

 p the perpendicular from the planet upon asymptote to C ; 

 a the acute angle between the transverse axis of C and the 



asymptote to C ; 

 <^ the angle between the tangent to 3 at O (drawn in the 



direction of the planet's motion) and the line from the 



planet to the vertices and centre of C ; 

 A the semi- trans verse axis of C ; 

 B the semi-conjugate axis of C (hence equal to _p) ; 

 r the distance of the planet from the sun ; 

 Tf the distance of the comet from the sun ; 

 To the distance of the comet from the planet ; 

 Pi and p distances of the comet from the sun at selected epochs before 



and after perturbation ; 

 Ui and u the velocities of the comet at the selected epochs ; 



A the increase to which v"^ — -•' - — . — ■- — receives by the planet's 



r. To 



action during the whole period in which the comet is 



passing near to Jupiter. 



