266 Prof. Challis on Hydrodynamics. 



first order, and a- l is consequently expressed by a circular func- 

 tion having as many plus as minus values, the pressure accom- 

 panying the condensation a has no tendency to produce a perma- 

 nent motion of translation of the sphere. Let us next inquire 

 whether such motion is indicated when terms of the second order- 

 are taken into account. 



As we have hitherto employed only the first terms of the 

 values of acr 1 and w f , the differential equations we have been 

 concerned with are linear with constant coefficients, and from 

 such equations the coexistence of small vibrations has been in- 

 ferred. But it is particularly to be observed that the general 

 composite character of the vibrations of an elastic fluid is not de- 

 monstrated by means of these approximate equations, inasmuch 

 as the proper proof consists in finding antecedently, and without 

 reference to arbitrary conditions, expressions giving the laws of 

 simple vibrations. On this account it is allowable to assert 

 generally that the motion is compounded of motions defined by 

 any number of terms of the series for aa l and w'. Since m is 

 supposed to be very small compared with a, the convergence of 

 each series is very rapid, and the terms following the first are 

 collectively very small compared with the first. Also, as has 

 been already urged, the vibrations, whether great or small, of a 

 given particle through, a mean position necessarily require that 

 the condensations should be greater than the rarefactions, which 

 accords with what is indicated by the equations (a) and (/3). For 

 these reasons I conclude that the motions and condensations 

 represented by the additional terms always accompany those ex- 

 pressed by the first; and that in composite motion each compo- 

 nent is in strictness defined by all the terms of the series for aa' 

 and w', excepting so far as modifications are produced by the 

 transverse vibrations. 



It is, however, true that after advancing beyond terms of the 

 first order, the reasoning no longer rests exclusively on inferences 

 drawn from linear differential equations with constant coefficients. 

 Yet, as I now proceed to show, certain results, sufficient for the 

 present purpose, may be obtained when terms of the second 

 order are included in the reasoning. First, it is to be noticed 

 that w' is periodic in such manner as to have as many plus as 

 minus values, whatever number of terms be taken, although this 

 is not the case with respect to the corresponding value of a 1 . 

 Also as the composite velocity is by supposition wholly periodic, 

 it, like the components, must have equal amounts of plus and 

 minus values. To satisfy this condition, it clearly suffices to 

 suppose that the composite velocity is the sum, or proportional 

 to the sum, of the individual velocities. Let, therefore, the total 

 motion be composed of an unlimited number of separate vibra- 



