18 RELATIONS PERTAINING SIMPLY [BOOK I. 



The radius vector r is not fully determined by v and (f, or by Jf and 9, but 

 depends, besides these, upon p or ; its differential, therefore, will consist of three 

 parts. By differentiating equation II. of article 8, we obtain 



d r d p . e sin v -, cos m cos v , 



= -4-- -dfl =-r-2- -dm. 



r /&amp;gt; 1 -)- e cos v 1 -[- e cos v 7 



By putting here 



Ap da , , 



- = -- U tan (f d q&amp;gt; 



p a 



(which follows from the differentiation of equation I.), and expressing, in con 

 formity with the preceding article, d v by means of d M and d y, we have, after 

 making the proper reductions, 



dr da , a , -, -,. a , 



== -- 1 tan (p sin vd M -- cosy cos v dtp, 



dr = - da -f- a tan y sinvd M a cosy coswdy. 



Finally, these formulas, as well as those which we developed in the preceding 

 article, rest upon the supposition that v, (f, and M, or rather d v, d (p, and d M, 

 are expressed in parts of the radius. If, therefore, we choose to express the vari 

 ations of the angles v, (p, and M, in seconds, we must either divide those parts of 

 the formulas which contain d v, d 9, or d M, by 206264.8, or multiply those which 

 contain dr, dp, da, by the same number. Consequently, the formulas of the pre 

 ceding article, which in this respect are homogeneous, will require no change. 



17. 



It will be satisfactory to add a few words concerning the investigation of the 

 greatest equation of the centre. In the first place, it is evident in itself that the dif 

 ference between the eccentric and mean anomaly is a maximum for E= 90, 

 where it becomes = e (expressed in degrees, etc.) ; the radius vector at this point 

 = a, whence v = 90 -j- &amp;lt;jp, and thus the whole equation of the centre = (p -(- e, 



