Olbers's Essay on Comets. 143 



(5) Log M 9.940836 



+ Log tang j3' 9.508173 



+ Log tang 0" 9.643090 



= Log .12362 9.092099 



Then adding twice each of these numbers to r'^+r'"'^, we have 

 r'-' + /"^=: 2.02464— 1.50336 /+ 2.0 1262 §'* 

 —2.00596 + 1.39382/- 1.51560 g'^ 



k"''= .01868- .10954^'+ .49702 g'^ 



Our three equations therefore become 



^"—^(1.01011— 1.21482 /+ .90869 /^) 

 ■/•'=V (1-01453— .28854 /+ 1.10393 /^) 

 r=V( .01868- .10954/+ .49702/^). 

 In which if we take ^'==1, we have r'::=1.40..., r"'=r.84..., 

 and A"=.62..., giving the time in which the area is described 

 26.88 days; while the observation gives only 8 : consequently 

 this value of / is much too great. 



We will therefore take r=:.5 ; then r' will be 1.07, r"'=r.80, 

 and ^"=.297, and the time 11.83 days, which is still too great. 

 We may next try /— ^; giving r'=:1.02, r"'=.84, ^"=.194, and 

 the time 7.79 days, or somewhat too small. Hence it may be 

 inferred that the true value of/ cannot differ much from .35. 

 We may therefore compute more accurately for .345 and .350, 

 and we shall find 



/=.345 /:^350 



?•'= 1.02294 »•'= 1.02409 



r"'= .83616 r"'==:. 83441 



k"z= .20012 r=. 20304 



T=: 7.9271 T=8.0410 



Consequently the error of the former supposition is — .0729, 

 of the latter +.0410; whence the true value of / is .34820; 

 and by an easy interpolation we find r'= 1.02367, 3-"'=. 83504 ; 

 and Log /"=Log M/i=9.482665. 



§ 47. 



In order to determine the whole orbit, we obtain the heliocen- 

 tric latitudes from the formula. 



