334 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



conical surface at the time t + r is m z , ( =—I ) or /» 2 . ( 1 + — 



V r ± nx J V r ± ar 



a Jm 



and this, neglecting terms of the order Sr x a-r, is equal to — - . 



Again, let v and p be the velocity and density of the fluid which passed 

 the area m 2 at the time /, and v /t p , the values of the same quantities at any 

 time t + t. Now the quantity of fluid which in the small time dt passes m % 

 18 equal to jm i p l v i dT, the integral being taken from t = to t = St. And 



because 



d.pv , 



pv t = pv + ~~jr- ~r very nearly, 



this integral is equal to 



, * , d.pv ^ty 



nfpvSt + m- . — ~ . ±~ + &c. 



Also if v], p[ be the velocity and density of the fluid which is passing the 



area — — of the exterior surface at the time t + t, the quantity of fluid 

 r 



— j— vlp'dr, taken from r = to r = St. 



And, because v] and p,' are what v and p become by very small changes of 

 time and place, 



, , d.vp d.vp* 



Hence, 



/•jbV" , , 7 m 2 / 4 r . d.vp d.vp, .' 



r 4 l r dt 2 dr 



Consequently, supposing the velocity positive when directed from the 

 centre, the increment of matter in the space between the two areas in the 

 time St, is ultimately, 



~ ^r^ v P + C -dT lr) ~ "* v p}* e '> 



