8 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



magnitude are taken account of. A comet is treated thiougbout tliis papei' as a small indivisible 

 body wliose mass may be neglected. 



i. Notation. — The symbols used in (1) and also other symbols wliicli I shall ha\- -casiuu to 



use may be thus defined : 



Let !& =orbit of the comet about the sun before tlie comet comes under the appreciable action of 

 the planet; 



(fi =orbit of the comet about the sun after perturbation by the planet; 



C =:hyperbolic orbit of the comet relative to Jupiter when near the planet; 



^ =elliptic orbit of Jupiter about the sun; 



A =point on (ff, which is nearest to ^ ; 



E =point on ^ which is nearest to (f, ; 



d =length of the straight line EA being the perpendicular distance between the orbits at their 

 nearest approach ; 



(.; =raugle between the tangent of (C, at A and the tangent to |( at E ; 



h =distance which the planet has yet to i)ass over to reach E when the comet is at A {h may 

 be negative); 



ni =niass of the planet, sun's mass=unity; 



a =unit of distance, in general the mean distance of the earth from the sun ; 



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



«, =planet's velocity in its orbit at E; 



,j,^ —comet's velocity in its orbit C when the comet enters t^e sphere of Jupiter's perceptible 

 influence; 



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



s =ro/*'/; 



/w, =semiaxis major of ®, (negative if ®, is an hyiierbola) ; 



® =semiaxis major of Oi; (negative if ® is an hyperbola); 



p ^perpendicular from the planet upon asymptote to C ; 



a =acute angle between the transverse axis of C and the asymptote to C. 



(p =angle between the tangent to il at O (drawn in the direction of the planet's motion) and 

 the line from the planet to the vertices and center of Cj 



A =semitransverse axis of C; 



B =semiconjugate axis of C (hence equal to p) ; 



r =distance of the planet from the sun ; 



r, =distance of the comet from the sun ; 



Tq =distance of the comet from the planet ; 

 p, and /j=distances of the comet from the sun at selected epochs before and after perturbation ; 

 M, and M=velocities of the comet at the selected epochs; 



A =increase which v^—— -^ — receives by the planet's action during the whole period 



in which the comet is passing near to Jupiter. 



5. If we assume two epochs, one before and one after the perturbation, at which the comet is 

 eipially distant fi-om the planet, the term 2mfa^/rQ is the same at both instants, and it disappears 

 from the value of A. Therefore 



