Prof. Challis on the Dispersion of Light 491 



Hence, by substituting in the equation (a), 



At the same time the equation of constancy of mass to the same 

 approximation is 



da iU 2U dW W L£S n , s 



-T.+-T + — +- ^z + — cot0 = O 7 



at dr r rdv r 



Now, if it be required to apply the equations (/3) and (7) to 



cases of motion in which there is no other relation between U 



and W than that which results from the mutual action of the 



parts of the fluid, since the analysis is required to determine 



that relation, we must equate separately to zero the quantities 



in brackets in the equation (/3). In fact, if that equation be 



multiplied by St, it will seem to be formed by a combination of 



D'Alembert's Principle with the Principle of Virtual Velocities ; 



and as by hypothesis there is no given relation between the 



virtual motions V8t and W$t, the factors by which they are 



multiplied must separately vanish. If U and W be eliminated 



from the two equations thus obtained and the equation (7), the 



result is 



1 d^.ar d 2 .ar Ifd^.ard.ar , „\ ,~ x 



This equation has been employed to determine the resistance of 

 the air to a vibrating sphere, the centre of the sphere being the 

 origin of coordinates. It appears, however, to be only applicable 

 to the case in which the fluid is confined within fixed boundaries. 

 For it is evident that in that case just as much incompressible 

 fluid must flow in the direction contrary to the motion of the 

 sphere as the volume of the sphere displaces ; which is precisely 

 the result to which the analysis conducts when the vibrations 

 are not very rapid 3 and the movement of the fluid is consequently 

 very nearly the same as if it were incompressible. The analysis 

 also shows that the same equation applies to the case in which 

 the sphere is fixed and the mass of fluid is caused to vibrate 

 bodily. In fact we may pass from the one case to the other by 

 conceiving motion equal and opposite to that of the sphere to be 

 impressed both on the sphere and on the fluid, such impressed 

 velocity not altering the value of a, and therefore not altering 

 the function that a is of ?*, 0, and t } as given by the above 

 equation. In both these cases the velocities U and W are related 

 to each other in a manner depending only on the mutual action 

 of the parts of the fluids, the fixed boundaries being supposed to 

 be so far distant from the sphere as not to affect the law of 

 the motion. 



2K2 



