SECT. 3.] PLACES IN ORBIT. 147 



both angles seem to admit of a double, and the resulting time, of a quadruple, 

 determination. We have, however, from equation 5, article 88, 



cos/. \l rr 1 =. a (cosy cos h] = 2 a sin d sin e : 



now, sin & e is of necessity a positive quantity, whence we conclude, that cos/ 

 and sin i $ are necessarily affected by the same sign ; and, for this reason, that 

 d is to be taken between and 180, or between 1 80 and according as cos/ 

 happens to be positive or negative, that is, according as the heliocentric motion 

 hap pens to be less or more than 180. Moreover, it is evident that d must neces 

 sarily be 0, for 2/= 180. In this manner d is completely determined. But 

 the determination of the angle continues, of necessity, doubtful, so that two 

 values are obtained for the time, of which it is impossible to determine the true 

 one, unless it is known from some other source. Finally, the reason of this 

 phenomenon is readily seen : for it is known that, through two given points, it 

 is possible to describe tivo different ellipses, both of which can have their focus 

 in the same given point and, at the same time, the same major semiaxis;* but 

 the motion from the first place to the second in these ellipses is manifestly per 

 formed in unequal times. 



107. 



Denoting by # any arc whatever between 180 and -|- 180, and by s the 

 sine of the arc $% , it is known that, 



Moreover, we have 



//I 1.1 5 1.1.3 



* sin % = s v/ (1 ss) = s i s 8 274 s 5 ^-j-g 

 and thus, 



* A circle being described from the first place, as a centre, with, the radius 2 a r, and another, 

 from the second place, with the radius 2 a /, it is manifest that the other focus of the ellipse lies in the 

 intersection of these circles. Wherefore, since, generally speaking, two intersections are given, two dif 

 ferent ellipses will be produced. 



