NEW FORMULA RELATIVE TO COMETS. 



279 



For this purpose, equate the two values of x, — X,, which result 

 from the expressions preceding (5) ; and from these and analogy we 

 obtain 





(" 



> 



(T) 



in which we have put i = ^" — f, yj = ■/}" — yj'. 



Resolve these equations relatively to p, p, p", with their coefficients; 

 and in order to exhibit the results in brief terms, let the coefficients 

 which are found to affect |, >7 be denoted by 



A = ^y'-/3y, A' = /3V— /?/, A" = (3y—(3'r ', ) ... 



= a y — ay , x$ = ay — a y, Jti = ay— ay. \ 



These are known from observation of this comet at C, C, C". They 

 give the conditional equations 



A/? -I- M^' 4- A"/?" = 0, Ay -j- Ay H- A'y = 0, ^ 

 Ba + B'a' -f B'a" = 0, By -}- B/ -f- B'y = J ^ 



and enable us to express the values now under consideration thus : 



(^ . Ap-A^+B, 1 A-^+B-, 1 ._ A-^+B-, . 



W^«V^ D ' 1)'^ D ' V^ D ' ^^^ 



the common denominator being either of the forms 



D = Aa 4- Ma! + A'^a" = B/3 -f- B'/3' + B''/?'' ; (1 1) 



which and (9) will be of immediate use in the simplification of for- 

 mulae (6). 



VI. — 3 u 



(9) 



