SECT. 3.] PLACES IN ORBIT. Ill 



, 2 COS 2 i A COS 2 2 O) , 8 



= a-\-~ j = a-\ -- 5, 



2/1 &amp;lt;-&amp;gt;\2 /I \2 



cos 2 o(l -- r (1 -- r 



\ pi \ p&amp;gt; 



if we put 



2 /cos^^coV e _ 



\ COS &amp;lt;U / 



Writing, therefore, n for y/jt?, 7t will be determined by the equation 



which properly developed would ascend to the fifth degree. We may put 

 TT =: q -\- p, so that q is the approximate value of n, and ^i a very small quantity, 

 the square and higher powers of which may be neglected : from which substitu 

 tion proceeds 



or 



and so 



_ 



(qq-S) 



Now we have in our problem the approximate value of n, namely, 3 a, which 

 being substituted in the preceding formula for q, the corrected value becomes 



_243ce 4 -}-3(9tt 3) (9ce-{-73) 

 (9a 3) (27 ace + 5 3) 



Putting, therefore, 



27~o^ = P&amp;gt; (1 3p)a y &amp;gt; 

 the formula assumes this form, 



n = 



and all the operations necessary to the solution of the problem are comprehended 

 in these five formulas : 



I. - = tan (45 + 01) 



