50 RELATIONS PERTAINING SIMPLY [BOOK I. 



We present the equation between the time t and the auxiliary quantity u in . 

 the following form : 



in which the logarithms are hyperbolic, and 



2V(~ 



is a quantity of the first order, 



J( 5 



a quantity of the third order, when log u may be considered as a small quantity 

 of the first order. Putting, therefore, 



i) + ^log 



n .1 A ~ &quot; 



^ 



A will be a quantity of the second order, but B will differ from unity by a differ 

 ence of the fourth order. Our equation will then assume the following form : 



..... [2] 



which is entirely analogous to equation [1] of article 37. Putting moreover, 



T will be a quantity of the second order, and by the method of infinite series 

 will be found 



Wherefore, putting 



C will be a quantity of the fourth order, and 



A 



Finally, for the radius vector, there readily follows from equation VII., article 21, 



? _ 



(1 T)co^i v ~~ (l-Tpl-j- C)cos*$v 



