SECT. 3.] PLACES IN ORBIT. 105 



of IT must be already known. After IT is found, p will be computed by the 

 formulas 



p r (1 + cos (N IT)) = r (1 -f- e cos (N 77)), 



or by this, 



_ 2 r/ e sin | (N N) sin ( N -{^ N 11) 







82. 



Finally, let us suppose that there are given three radii vectores r, r, r&quot;, which 

 make, with the right line drawn from the sun in the plane of the orbit at pleasure, 

 the angles N, N , N&quot;. We shall have, accordingly, the remaining symbols being 

 retained, the equations 



(I.) = 1 -f e cos (N 77) 



. l-|_ ec os(iV 77) 



2r=l + ecoa(N&quot; 77), 



from which p, 77, e, can be derived in several different ways. If we wish to 

 compute the quantity p before the rest, the three equations (I.) may be multiplied 

 respectively by sin (N&quot;- -N \ -- sin (N&quot; -N), sin (N -N\ and the products 

 being added, we have by lemma I, article 78, 



sin (N&quot; N ) sin ( N&quot; N) + sin (N r N) 

 i sin (N&quot;N ) - - ^ sin (N&quot; N) -f ~ sin (N* N} 



This expression deserves to be considered more closely. The numerator evidently 

 becomes 



2 sin k (N&quot; N } cos i (N&quot; N ) 2 sin } (N&quot; N } cos ( } N&quot; + I N N) 

 = 4 sin * (N&quot; N } sin * (N&quot; N) sin * (^ JV). 

 Putting, moreover, 



/ r&quot; sin (JT N ) = n,r r&quot; sin (iV&quot; JV) = n , r / sin (JT JV) = &quot;, 



it is evident that i n, k ri & n&quot;, are areas of triangles between the second and third 

 radius vector, between the first and third, and between the first and second. 



14 



