Prof. Challis on Hydrodynamics. 267 



tions, direct and transverse, having their axes perpendicular to a 

 plane ; and let the direct vibrations be expressed by two terms 

 of the series for w 1 , and their phases be such that the transverse 

 vibrations destroy each other. The total motion is consequently 

 in parallel straight lines ; and the rate of propagation being tea, 

 the composite velocity (V) and density (p) are related to each 

 other by the general formula Vp = tca(p — 1), obtained in Part I. 

 (p. 218), which must now be used to terms of the second order. 

 Hence if S represent the composite condensation, we shall have 



*aS = V+ — , 

 /ca 



in which, from what is said above, V is either S . w' or proportional 

 to this sum. From the form of the expression for iv', it follows 

 that (2 . w'y 2 is equal to the sum of the squares of periodic terms 

 together with periodic terms having equal amounts of plus and 

 minus values. Hence it appears that part of the value of S is 

 always positive. 



Reverting now to our problem of waves impinging on a sphere 

 at rest, to proceed to the second approximation we have to em- 

 ploy S in the place of a^ for the condensation of the waves. But 

 since S differs from a l by a very small quantity, if we call /(a-,) 

 the first approximation to a, we shall have very nearly for the 

 second approximation, 



Now the value of/^) found above is of the form 0^(1— wijQ), 

 Q being independent of a v Hence by the above formula 

 o- = S(l — mjQ) ; that is, to advance to the second approxima- 

 tion, it suffices to substitute S for a l in the value of o\ Con- 

 sequently we have 



„ 87r 3 m,fcac ~ Q 



a— $ = -~ — . S cos 6. 



mAr 



The total pressure on the sphere tending to produce motion in 

 the direction of the propagation of the incident waves is 



27rjV(o— -S)c 2 sin6^cos<9dl9, from 6 = to 6=ir. 

 This quantity will be found to be 



_32ttWc 3 Sm, 

 3X 3 ' m ' 



Hence if A be the ratio of the density of the sphere to the den- 

 sity of the fluid, the accelerative force due to the action of the 



