NEW FORMULAE RELATIVE TO COMETS. 289 



ties X„ Y, ; and put ii'=l, (^"=1, &c. and also p = hMt't", there 

 will result for the component velocities x„ y„ z„ the less approxi- 

 mate values 



a;, = X, -I- A (t'M'a' — t"M."a" — (f — t") Diy 



y,=Y.-{-h {t'm!li' — t"M"^"- (t' — t") By!,), y (a) 



z,=h (/'My - f'M VO ; J 



and if to these be added the identical equations 



Ma -h M'a' -f M''a" — B^, = 0, 

 M^-\-M'^'-\-M"(3" — By!,=:0, 



My4-My-f-MV"=0, 



respectively multiplied by ^ h {t" — f), we shall immediately obtain 

 the following expressions : 



X, = X,-\-h{id(M'a' — M"a") — ia'(Ma + De)} 

 y. = Y^-\-h{ie (M'/3' - MT) -he' (M/3 ^ Byj,)] 

 z,z=h{ie (My— MY) — i a' My} ; 



> (&) 



in which 6 = t' -\-t", B' — t' — t". 



These are in fact less simple than the preceding. They become 

 identical with the values F,, G,, H,, found at page 44, vol. II. Theo- 

 rie Analytique, &c. when we further neglect the arc a in the values 

 of ^„ >7,, multiplied by the factor D. The term B^, may be removed by 

 directing the axis of y to the earth when the comet is at C. The 

 preceding values oi x„ y will then agree with P,, Q,, page 45. But 

 even in this simplified state, the computation depends on nine difTerent 

 angles instead of the three involved in formulae (A). 



The preceding values (a) and (b) are not the only forms that can 

 be derived from (19), in virtue of the identical equations which we 

 have just employed. If to (a) we add those equations respectively 



