278 NEW FORMULA RELATIVE TO COMETS. 



TT/ 1 JL ' _X-i —Li-' &ro • V /' ^ -L 1 ^ ^p 



The forms for x'\ X" in (2) are similar to these, observing to change 

 the single into double accents, as respects w, U ; ?^, V ; and also t' into 

 — t" ', and like expressions v^^ill evidently apply in the directions of the 

 axes of 1/, z. 



Substitute (3) and the similar forms for x% X" in (2), and then eli- 

 minate X by the first of equations (1). There will result the tvi^o ex- 

 pressions : 



v'{x,^X,) = u'^a—9'a'J^v'e, ?)" (a:,— X,) = p"a"— M"pa-|- v''^', 

 in which, for brevity, we have assumed ; 



v'k' — (V— u')X,— (U'— M')X, vT = (V"—v")X,-{-(V"—u")X (5) 



v" V 



and which, being respectively multiplied by — , — and added, will 

 give 



(x,-X,)(»'+t.")=(^, v'—~„ V )(■«--, pV + -, fV+v'f+vV 



the two last being formed by analogy from the first. 



Expressions apparently more simple than these, might have been 

 found from the preceding values ; but the present forms are more con- 

 venient for the determination of the component velocities x,,y„z„ which 

 will become known by means of the earth's velocities X„ Y,, when 

 we have expressed the geocentric distances p, p', p', in terms given by 

 observation of the comet in the corresponding positions C, C, C '. 



