148 RELATIONS BETWEEN SEVERAL [BOOK I. 







We substitute in this series for a, successively 



__ 



and we multiply the results by a 2 ; and thus obtain respectively, the series, 



l (r + /_^ + ^l (r + /_^ + T ^J-(, + /_^ + 



T?ii* ( r + r e) 1 + etc. 



the sums of which we will denote by I 7 , 27 Now it is easily seen, since 



the upper or lower sign having effect according as 2/ is less or more than 180, 

 that 



a *(d sm8) = F, 



the sign being similarly determined. In the same manner, if for e is taken the 

 smaller value, inferior to 180, we have 



a (e sin e) = 17; 



but the other value, which is the complement of the former to 360, being taken, 

 we evidently have 



a* (e sin e) = a* 360 U. 



Hence, therefore, are obtained two values for the time t, 



a$360 U+T 



108. 



If the parabola is regarded as an ellipse, of which the major axis is infinitely 

 great, the expression for the time, found in the preceding article, passes into 



