328 Prof. Challis on a Mathematical Theory 



in this case tlie same as that given by the solution of Problem III., 

 at least to the first order of small quantities. But in the circum- 

 stances of Problem IV., dififerent parts of the fluid move with 

 different velocities, the motion being accompanied by condensa- 

 tion or rarefaction. As the excursions of the vibrating particles 

 of the fluid are supposed to be very large compared with the 

 diameter of the sphere, this difference of ciroimstance is taken 

 into account with sufficient approximation by the factor pj, as is 

 shown by the foregoing investigation, so far at least as relates to 

 the hemispherical surface on which the waves are incident. With 

 respect to the other part of the surface, other considerations have 

 to be taken into account, as will appear from the following 

 argument : — 



The value of a-[s — a) found above, determines completely the 

 pressure on the surface immediately acted upon by the waves, 

 because the velocity W sin 6 along that surface is impressed at 

 each instant by the waves themselves. But the disturbance of 

 the fluid in contact with the other hemisphere results, not from 

 the direct action of the waves, but from the momentum of the 

 fluid in the plane through the centre of the sphere perpendicular 

 to the direction of their incidence. Now it may be asserted ge- 

 nerally, that the sum of the pressures produced on the surface of 

 the sphere in a given time is proportional to the sum of the mo- 

 menta of the fluid which passes the above-mentioned plane in 

 the same time. But if during the transit of the condensed por- 

 tion of a wave across the plane the sum of the momenta be 

 2 . MW(1 + a), during the transit of the rarefied portion the sum 

 of the momenta will be 2 . MW(1 — o-), taking effect in the con- 

 trary direction. The former of these quantities exceeds the other 

 by 22 . MWo-), which varies as W^, because a varies asW. Hence 

 there will be a residual pressure, varying as the square of the 

 velocity of the fluid, and acting in the direction contrary to that 

 of the propagation of the waves. This important inference ad- 

 mits of the following more particular investigation. 



In the solution of Problem II., reasons were given for con- 

 cluding that, in the case of a uniform stream, the velocity of the 

 fluid in contact with the sphere is W sin 6 for both hemispheres. 

 In the case of incident waves, the velocity has this same value 

 for the first hemisphere, and, from what has been argued above, 

 approximately this value for the surface now under consideration. 

 To obtain the complete value for the latter, it must be considered 

 that the disturbance is propat/aicd from the plane perpendicular 

 to AA' through the centre of the sphere, and that the velocity at 

 A' is zero. Hence, in consequence of the confluence of the lines 

 of motion along the surface, there will be at this point a partial 

 reflexion of velocity, so much the greater as the curvature of the 



