20 Proceedings of the Royal Irish Academy. 



If the integral is independent of tlie mode of passage, tlie con- 

 dition (46) is 



^Si?-SvV^> = 0, (73) 



Of 



and the simplest physical illustration seems to be to take ^p = c to 

 he the density of a fluid and Y^ = co- to he the ^product of the 

 density and the velocity. The equation of continuity being 



|;-SV(.cr) = 0, (74) 



the condition is satisfied, and the integral 



Q = -I d'7 JJ <; S o- dp d'/3 +111^8 dp d'p d"p (75) 



is independent of the mode of passage. 



The integral JJ c S o- dp d'p is the flux of the fluid through the 

 surface with which the variable instantaneous surface momentarily 

 cbincides, and the integral - J d"# JJ c S o- dp d'p is the negative of 

 the time integral of this flux corresponding to the motion of the 

 instantaneous surface. The integral ^ c S dp d'p d"p is simply the 

 negative of the quantity of fluid which has passed through the 

 instantaneous surface in its motion. 



