Central Forces. 49 



By substituting in (5) k)v p' its value as given by (7), it reduces 



V'2 sin. =4.— J ^ ^ 

 to COS. ■y= (9)5 at the perihelion cos. y — 1, sin.4y= 1, 



.'•e=y7^ — 1 (10); hence if V and V are known at the perihelion, 



the conic section is given in species, for e=the eccentricity di- 

 vided by the semi-axis; at the aphelion cos. y=.l, sin. 4^ = 1, hence 



e=l — y77 (11); -''by knowing V and V at the aphelion the conic 



section is given in species as before. If the particle should receive 

 a new velocity at any given point of its course, by means of an im- 

 pulse which acts in a direction that makes given angles with r and 

 the tangent to its course at the poiat of impulse; then by compound- 

 ing the velocity of the particle in its orbit at the moment of impulse 

 with the velocity of impulse, the new velocity with wiiich the parti- 

 cle will move becomes known both in magnitude and direction; 

 hence s]^, the angle made by r, and the direction of the velocity thus 

 found becomes known, also V remains the same after the impulse 

 as before; hence substitute in (7) for ■\j and V the new angle and 

 velocity found as stated above, and p', the new semi-parameter be- 



r 

 comes known ; also - becomes known by substituting in (6) for V 



the new velocity, .' .since r is known a is found; hence a and p' be- 

 ing found the conic section becomes known, hence e is found also; 

 then substituting in (9) for e the value thus found, and for 4- and V 

 their values found as above directed, and cos. v is found, which gives 

 the position of the perihehon of the new orbit. 



The inclination of the new orbit to the orbit described before the 

 impulse, is easily found by spherical trigonometry ; their node line is ev- 

 idently the line r drawn from the centre of force to the point of impulse, 

 (See Prin. sec. iii. prop. 17, cor. 3, and Mec. Anal. Vol. II. p. G6.) 



Also, should the particle receive a succession of impulses (acting 

 in given directions) as it moves; then by proceeding in the same man- 

 ner, by compounding every new velocity with the velocity at the mo- 

 ment of impulse, the curve described becomes known, as stated in 

 cor. 4, prop. 17. 



If the impulses act continually, then the integral calculus must be 

 employed according to the methods pui'sued in estimating the dis- 

 turbing forces of the planets. (See Mec. Anal. Vol. II. p. 76.) 



Vol. XIX.— No, 1. T 



