284 NEW FORMULA RELATIVE TO COMETS. 



scosa ='R—^{t' — t")p, ssma = ^{f — t")q', 

 from which there will result 



3R — (r — t)p 3 sin a ^ ^ 



and we shall have instead of (17), the values 



^, =scos(L — a)j *7, =ssin(L — a); 



in which a is evidently a small arc and s nearly equal to R. 



A further simplification of the expressions last given, may be effect- 

 ed without diminishing their accuracy, or rendering more complex the 

 velocities X,, Y,. Conceive the axis of a;, the position of which is 

 arbitrary, to be directed so as to form an angle equal to a with the ra- 

 dius vector R. From this position which we suppose to be less ad- 

 vanced than R, the angle L — a will then take its origin, and we shall 

 have 



£=5, >7, = 0; X = Rcosa, Y = Rsina; (22) 



the corresponding velocities being 



X, = J9 cos a — qsma^l 

 Y, =^ psma-\- q cos a, ) 



(23) 



in which the first term of Y, is extremely small. 



With respect to the angular quantities in the equations (1), now to 

 be considered ; let / denote the geocentric longitude of the comet in its 

 position C ; and x the corresponding geocentric latitude. If we also 

 represent by p the curtate distance of the comet's projected place in 

 the plane of the ecliptic, we shall have, in the present position of the 

 axis of ic, the following values : 



a = cos(a-f-^ — L), /(? = sin(aH-/ — L), y^ tan Pi, (24) 



and the co-ordinates (1), by adding their squares, will give the equa- 

 tion: 



