196 Mr CHALLIS's RESEARCHES IN THE THEORY 



Let now the area of eg = m, and that of ac = m'. Then if the motion 

 which exists at a given instant, be supposed to be continued uniform 

 for the small time t, the quantity of fluid which passes the section eg 

 in that time, is mpF^T, and that which passes ac is m'p'Vr. Hence 

 the increment of matter between the two sections is — {m' p'V'T — mpVT), 

 whether the velocity tend from or to the focal lines, being considered 

 negative in the latter case. The increment of density {Ip) of the element 



in the time t, is consequently — ^^ — - — r—, — —. — — ■ But — = — ^^ =^ . 



^ •' m{r'-r) m r{r + l) 



Hence 



pT'r'ir' + D-prnr + l) _^^^^^^Sp_^ 



And passing from differences to differentials, 

 ^^^^^'dt ~ dr 



or 



dp dV ,.dp ,^ /i 1 \ 



As before udx + vdy + wd% = V dx + V^^ — — dy + V d% = Vdr, 



" f* T T 



if a, /3, 7, be the co-ordinates of the middle point of the focal line hi. 

 Now as we have supposed that in passing from one point to another 

 of tlie element acge, the change of velocity at a given instant depends 

 only on the change of r, we may consider V a function of r and t, 

 and Vdr a differential of a fimction of r and t. Then udx ^ vdy 

 + wdfi = d(l), a complete differential of a function of x, y, and as; and 



-~ = V. But in this case we have the known equation, 



a' Nap. log. p^fiXdx + Vdy + Zd%) - ^ - ~ (B.) 



Therefore considering X, V, Z, to be independent of the time, 



d'dp _ d^(p dV d'(ji d(p d'<p 



pdt ~ df dt ~d¥ ~ 'dr ' drdt ' 



