THIRD SECTION. 



RELATIONS BETWEEN SEVERAL PLACES IN ORBIT. 



78. 



THE discussion of the relations of two or more places of a heavenly body in 

 its orbit as well as in space, furnishes an abundance of elegant propositions, such 

 as might easily fill an entire volume. But our plan does not extend so far as to 

 exhaust this fruitful subject, but chiefly so far as to supply abundant facilities for 

 the solution of the great problem of the determination of unknown orbits from 

 observations : wherefore, neglecting whatever might be too remote from our pur 

 pose, we will the more carefully develop every thing that can in any manner 

 conduce to it. We will preface these inquiries with some trigonometrical propo 

 sitions, to which, since they are more commonly used, it is necessary more fre 

 quently to recur. 



I. Denoting by A, B, C, any angles whatever, we have 



sin A sin ( C B} -f- sin B sin (A C} -\- sin (7 sin (B A) = 

 cos^sin ( C B} -|- cosB sin (A C) -f- cos Csm(B A) = 0. 



IT. If two quantities p, P, are to be determined by equations such as 



psin(A P) = a 

 psan(BP) = b, 

 it may generally be done by means of the formulas 



p sin (B A) sin (H P} = b sin (H A} a sin (H B} 

 p sin (B A) cos (H P) = b cos (// A) a cos (IT B), 



in which If is an arbitrary angle. Hence are derived (article 14, II.) the angle 

 H P, and p sin (B A) ; and hence P and p. The condition added is gen- 

 (100) 



