SECT. 1.] TO POSITION IN THE ORBIT. 29 



Hence, by eliminating d u by means of the preceding equation we obtain 



djST rr -, . /-, . r\ r sin v -, 



-r- = TTT d v 4- ( 1 -\ I? d w , 



X. ootanifi \ p/OOOBIfi 



or 



dani , , T /b . b \ sin v tan T/I 

 V = T --dJV (- - 







r p / cosii&amp;gt; 

 JJtaniOj ,, /-. . \sint- , 



= Y - d iv ( 1 4- - ) - - d w , 



t.rr \ r/smty 



28. 



By differentiating equation X., all the quantities r, b, e, u, being regarded as 

 variables, by substituting 



dsnil/ -, 

 e = f- dw, 



cos 



and eliminating dz with the help of the equation between dJV, d, dif, given in 

 the preceding article, there results, 



r , , , l&amp;gt;bt&amp;gt;,(uu 1) , , r . b ( , 1. . , 1\ ) n 



^i;d6-] day -4- 5- j- &amp;lt; (M + -) smw (u -- ) sin v } aw. 



b 2iur I 2cos-i \ u v u i 



The coefficient of d N is transformed, by means of equation VIII., into , ~ : but 



J I sm i/) 



the coefficient of d y, by substituting from equation IV., 



u (sin y siny) = sin (y v}, - (sin if -(- sin y) = sin (i// -f- f ), 

 is changed into 



5 sin i/; cos v __ p cos u ^ 

 cos 2 1// sin i/&amp;gt; 



so that we have 



6 ?. sin i/; 



So far, moreover, as N is considered a function of b and t, we have 



which value being substituted, we shall have d r, and also d v in the preceding 

 article, expressed by means of d t, d b, d t//. Finally, we have here to repeat our 



