302 CELESTIAL MECHANICS. I. vil. 35. 



The initial dimensions of an elementary rectangular pa- 

 rallelepiped being here Dr, rD-ar sin 6, and rD9, calling the 

 values ofr, 9, and -ar after the time t, r' , & and -ar', and fol- 

 lowing the steps of article 365, we shall find that the 

 volume of the elementary figure will become equal to a 



dr 

 rectangular parallelepiped, of which the height is --- or; 



the breadth r' sin & v - — D'sr-u--— Dr i» from which Dr may 

 d'zar dr / 



dr' dr 



be exterminated by the equation - — Dw-f -= — DrzzO; 



d'sr dr 



(dd' d^ dd' \ 



-T--dr+-^D5 + .j-D'arj, pro- 

 vided that we make 



dr' dr' dr' 



-^Dr+-^D5+-^D^=:0, and 



Q'sr dw d^jT 



-T — Dr-f- ■-TT-D54- --1 — D'arzzO; [r, 'ar and d, and r, 



-ar' and 6' being here substituted for a, &, c, ^, y and z] : and 



makin ^-^'^^ ii^4^_^il Jzl 

 ° "" dr'd^' d-ar dr ' d-ar' dd dQ ' d'sr ' dr 



dr' def d'ar' jd/ dfi^ d;^ dr' jd5^ d;nr^ 



dd ' dr ' d'ar d-ar dr ' dfl * d'ar ' d9 dr 



volume of the element, after the time t, will be Q' r'^ sin 5 

 Dr Dfl D'ar ; consequently, if we call the primitive density 

 (f), and f the density corresponding to the time f, we shall 

 have, since the masses must be equal, fCV^ sin d'=(f)r ^ sin 

 6, which is the equation of continuity ; and substituting for 

 r'fr-\-as\ for d', 5+ aw, and for-ar', nt+w-^av, we shall have, 



if we neglect the quantities of the order a^, C'zz \ +a 



-r- +a-TT+ * T- U^ tlie same manner as € was found equal 

 dr dfi d'ar "- ^ 



