NEW FORMULA RELATIVE TO COMETS. 283 



the several terms of which are completely determined as respects the 

 intervals t\ t". It remains therefore only to assign values to the earth's 

 velocities X,, Y, ; and to the equations (1). 



Denote by L the longitude of the earth when the comet is observed 

 in its mean position C. Let xs be the longitude of the perihelion of its 

 elliptic orbit, and e = sin e the eccentricity corresponding to the mean 

 distance 1. The value of of in 1801 was 99° 30' 5", annual in. = 

 -h r 2". At the same epoch, e was '01685301, sec. var. = 

 — •000041809. The co-ordinates of the earth in terms of the radius 

 vector R and longitude L, are 



X = R cos L, Y = R sin L ; 



and the known expressions for R and the elementary area described in 

 the instant dt^ are 



„ cose^ T^^dli 



R := ^-7Y ^v , R -77- = COS s. 



l-f-esm(L — ts)' dt 



If the three first of these expressions be differentiated relatively to 

 the time t of which R and L are functions ; and -j- be eliminated by 

 means of the last expression ; there will result 



X, = J9 cos L — q sin L, ) 

 Y, = ^ sin L H- g cos L ; ) 



in which we have assumed for brevity 



I. • /T \ COS£ ,^^. 



/) = tanesin(L — «) , q ^ -j^; (20) 



J.*, 



and which, in conjunction with the values of X, Y above given, will 

 make known the component velocities and position of the earth. 



The forms which we have here adopted, enable us to express the 

 values of ^„ >?, in simple terms. We may assume for the coefficients 

 of cos L, sin L in these values ; 

 VI.— 3 V 



