NEW FORMULAE RELATIYE TO COMETS. 



r^ = R2 _j_ 2Rp cos (l—h) 4- p^ sec r 



285 



(25) 



which establishes a relation between the curtate distance p and the co- 

 met's distance r from the centre of the sun. 



Expressions similar to these evidently apply to the extreme posi- 

 tions C, C" of the comet; and in (19), (25) we may, if requisite, 

 write p cos ?^ instead of p, which will then denote the ray drawn from 

 the centre E of the earth to the comet at C. 



When the distance r is given, we shall know p by means of (25) ; 

 and in such case all the quantities in (19) being determined, we may 

 compute the values of the required velocities x„ ?/„ z,. But should r 

 be also unknown as well as p ; and this is generally the case, the equa- 

 tion (25) will not alone be sufficient, and we must join to it the diffe- 

 rential expression 



which we adapt as a known form for the square of the comet's veloci- 

 ty in a parabolic orbit ; and which will enable us to obtain the dis- 

 tances r, p, and lastly the component velocities a;,, y„ z,. 



We may now present the expressions (19) in their final terms for 

 computation. They become, in terms of the simplified values of |„ 

 >7,, of the following form : 



X 



n f ll" ll' f /" 



■V -; J 



3s 







\ (A) 



in which the values of A, A', k." from (8) are 



A = ^y' — /3y, A' = /?V-/3/, A" = /3/-^V; 



and the corresponding value of D is by (11), 



D = Aa-f AV + AV. 

 3 v* 



