112 HELATIOXS BETWEEN SEVERAL [BoOK 1. 



IL __!_ = - 



Tn 2ein 2 J 4\/ (rr cos2iu) 



27 a a cos a&amp;gt; 



(1 3f?)cos 2 aj &quot; 



V. 



If we are willing to relinquish something of the precision of these formulas, it 

 will be possible to develop still more simple expressions. Thus, by making cos ia 

 and cos 2&amp;lt;o = 1, and developing the value of y/ p in a series proceeding according 

 to the powers of J, the fourth and higher powers being neglected, we have, 



in which J is to lie expressed in parts of the radius. Wherefore, by making 



Arr _ , , 



we have 



VI. = (l - 



In like manner, by developing &amp;lt;J p in a series proceeding according to the powers 

 of sin /I, putting 



we have 



VIL ^ p = 



or 



VTII. jo = / -f- $ sin 2 J\Jrr. 



The formulas VII. and VIII. agree with those which the illustrious EULER has 

 given in the Theoria motus plandarum ct cometamm, but formula VI., with that which 

 has been introduced in the Recherches et calculs sur la vraie orbite elliptigue de la 

 comete de 1769, p. 80. 



