SECT. 3.] PLACES IN ORBIT. 145 



log ...... 8.9308783 log ...... 8.7432480 



| log a ..... 0.9030928 flog a ..... 0.9030928 



aiogjfc ..... 1.7644186 C.log ..... 1.7644186 



logT ..... 1.5983897 lo S 2 ...... 0.3010300 . 



T= 39.66338 log* ...... 1.7117894 



t 51.49788 



Therefore, the perihelion passage is 13.91444 days distant from the time 

 corresponding to the first place, and 65.41232 days from the time corresponding 

 to the second place. Finally, we must attribute to the limited accuracy of the 

 tables, the small differences of the elements here obtained, from those, according 

 to which, the given places had been computed. 



106. 



In a treatise upon the most remarkable relations pertaining to the motion 

 of heavenly bodies in conic sections, we cannot pass over in silence the elegant 

 expression of the time by means of the major semiaxis, the sum r-\-r , and the 

 chord joining the two places. This formula appears to have been first discovered, 

 for the parabola, by the illustrious EULER, (Miscell. Berolin, T. VII. p. 20,) who 

 nevertheless subsequently neglected it, and did not extend it to the ellipse and 

 hyperbola : they are mistaken, therefore, who attribute the formula to the illus 

 trious LAMBERT, although the merit cannot be denied this geometer, of having 

 independently obtained this expression when buried in oblivion, and of having 

 extended it to the remaining conic sections. Although this subject is treated by 

 several geometers, still the careful reader will acknowledge that the following 

 explanation is not superfluous. We begin with the elliptic motion. 



We observe, in the first place, that the angle 2/ described about the sun 

 (article 88, from which we take also the other symbols) may be assumed to be 

 less than 360 ; for it is evident that if this angle is increased by 360, the time 

 is increased by one revolution, or 



-=aX 365.25 days. 

 19 



