124 RELATIONS BETWEEN SEVERAL [BOOK I. 



or both between similar multiples of the whole circumference, and also f and g 

 together, either between and 180, or between similar multiples of the semicir- 

 cumference. If, finally, the orbit should be wholly unknown, and it should not 

 appear whether the heavenly body, in passing from the first radius vector to the 

 second, had described a part only of a revolution or, in addition, one entire revo 

 lution, or several, our problem would sometimes admit several different solutions : 

 however, we do not dwell here on this case, which can rarely occur in practice. 



95. 



We pass to the second matter, that is, the determination of the elements from 

 the angle g when found. The major semiaxis is had here immediately by the 

 formulas 10, 10*, instead of which the following can also be used : 



rihr-i _2mmcosf^rr _ kktt 



1 / a - r~5 &quot; 



yysm g 



FT?*! 2MMcosf\/rr _ klctt 



~TTl\^~ ~4rrr/cos 2 



The minor semiaxis b = \/ap is got by means of equation 1, which being 

 combined with the preceding, there results 



Now the elliptic sector contained between two radii vectores and the elliptic arc 

 is bkt^p, also the triangle between the same radii vectores and the chord 

 irr sin 2/: wherefore, the ratio of the sector to the triangle is asy: 1 or Y: 1. 

 This remark is of the greatest importance, and elucidates in a beautiful manner 

 both the equations 12,12*: for it is apparent from this, that in equation 12 the 

 parts m, (l-{-x) 2 , X(l-\-x] , and in equation 12* the parts M, (L xf, X (L a?) , 

 are respectively proportional to the area of the sector (between the radii vectores 

 and the elliptic arc), the area of the triangle (between the radii vectores and the 

 chord), the area of the segment (between the arc and the chord), because the 

 first area is evidently equal to the sum or difference of the other two, accord 

 ing a.s v v lies between and 180, or between 180 and 360. In the case 



