46 Prof. Challis on the Hydrodynamical 



the fluid. We have next to calculate the quantity of fluid which, 

 according to these values of U andW, passes at any instant a given 

 transverse plane. 



These velocities, resolved parallel to CB, are respectively 



VR 3 VR 3 



— 3- cos 2 and ^-3 sin 2 0, so that the whole resolved velocity 



in that direction is 



VR 3 

 ^(3cos^-l). 



Hence the quantity of fluid which passes the part of the plane 

 exterior to the circle of radius y in the small time 67 is 



Stfeirr sin . ^(3 cos 2 0-l)d. r sin 0, 



the integral being taken from r = R to r = infinity. Since 

 r cos 0=x, this integral is equal to 



#£-»). 



ttVR 3 S/ 



which taken between the above limits is 



-7rVR 2 (l-|^ 2 V 



If the plane intersect the axis of the cylinder produced, at any point 

 beyond either A or B, we must suppose that y = Q, or that R 2 =# 2 . 

 Since in this case the integral vanishes, there is no permanent 

 transfer of fluid across such planes, with respect to which, there- 

 fore, the required condition is fulfilled. Thus the assumed ex- 

 pressions for U and W are so far justified. 



In other cases, by putting for R 2 the value y 2 + x* } the integral 

 becomes — 7rV?/ 2 . Now let f{x) be the mean velocity with which 

 the fluid within the distance y crosses the same transverse plane 

 in the direction from A towards B, then the whole quantity 

 that passes that plane in the time St is 



Trf{3c)tf L ht-->ttly*&. 



Since by the principle already enunciated this quantity is zero, 

 it follows that /(a?) =V. 



Hence, by having regard to the above signification of /(#), and 

 to the circumstance that the lines of motion converge towards 

 the parts about A and diverge from those about B, it is clear 

 that the velocity V diminishes with the distance from C accord- 

 ing to some unknown law. In default of an exact a priori in- 

 vestigation of this law, I shall now make the provisional suppo- 

 sition that V varies inversely as R 3 , or that VR 3 is equal to a 



