NEW FORMCL^ RELATIVE TO COMETS. 277 



tangular co-ordinates .r, y^ z. Refer the earth's centre E, to the axes 

 of X, 2/, supposed to be in the plane of the ecliptic, and let (X, Y) de- 

 termine its place at the distance R from S. The position of the comet 

 relatively to E, will then depend on the values of a; — X, y — Y, z ; 

 and if we denote its co-ordinates measured from E by pa, p/?, py, we 

 shall have in its position C 



a;=XH-pa, 2/ — Y-|-p/?, r = py (1) 



Accent the different letters in these expressions, in order that they 

 may correspond to two different positions of the comet at C, C" ; the 

 first being supposed to precede, the second to follow C, at the compa- 

 ratively small intervals of time /', t". The co-ordinates C, C", in the 

 direction of the axes of a?, and at the end of these intervals, will then be, 



a;' = X'4-p'a'; a:"=:X"H-pV; ' (2) 



and corresponding expressions will result in the directions of the other 

 axes of y and z. 



The determination of these co-ordinates in terms relative to the in- 

 termediate position C of the comet, and to the corresponding place E 

 of the earth, may be eflfected by M'Laurin's theorem, and the known 

 differentials 



37 „ X 



^" — —7-3' ^" — ~"Rl' 



which express the sun's attractive force on the comet and earth. By 

 means of that theorem, we have the expressions 



x'=x—x,t'-\-ixX—kxJ\ &c., ^=X—X,i+\Xj"—^XJ\hG. 



which, in virtue of the preceding differentials, take the usual form 



x' = v!x — v'x„ X'=U'X — V'X„ (3) 



the assumed coefficients of a:, x,^ and of X, X,, having in terms of the 

 interval /', the following values ; 



