40 Prof. Challis on Hydrodynamics. 



of the values of w for all the axes. V 1 is supposed to be pro- 

 portional, but not equal, to this sum, for reasons which will be 

 given presently. It is here assumed that any arbitrary func- 

 tion of z— icat-\-c may be expressed by the sum of an unlimited 



27T 



number of terms such as m sin — (z—icat -}-c), m, X, and c being 



at disposal. Again, if a 1 be the composite condensation corre- 

 sponding to V v and a the component condensation correspond- 

 ing to iv, by reason of the composition a l will be a function of 

 z — Kat + c, and the velocity of its propagation will be ica. But 

 the form of the function will not be exactly the same as that of 

 /, as will appear by comparing the two series for iv ' and g 1 . Also, 

 in consequence of the effect of the transverse motions, a 1 will 

 not be equal to 2) . a, but will be proportional to this sum. These 

 assertions rest on the following considerations. 



As by hypothesis the transverse motions are neutralized, and 

 as each set of vibrations has been proved to be independent of 

 the rest, it follows that the transverse motion relative to a given 

 axis is just counteracted by the effect of all the other transverse 

 motions. As far as regards the motion relative to that axis, the 

 effect is the same as if the fluid were impressed with extraneous 

 accelerative forces just equal and opposite to the transverse forces 

 relative to the axes which are due to the action of the fluid. Now 

 such impressed forces, being transverse, will not change the rate 

 of propagation, but will alter the relation of Vj to cr v That this 

 relation cannot be the same as that of w' to o ] will appear by con- 

 sidering that in the case of a single set of vibrations, the conden- 

 sations along the axis are partly due to the longitudinal accele- 

 rations and partly to the transverse accelerations, whereas in the 

 compound motion no part is due to transverse accelerations. We 

 might consequently expect that the condensation corresponding 

 to a given velocity would be less in the compound than in the 

 simple motion. That this is actually the case may be proved as 

 follows. Omitting in the series for w' and a' the terms involving 

 m 2 , m 3 , &c. as bearing in significantly on this question, we find 



that the velocity is equal to the product of - and the condensa- 



tion ; whilst in the case of the compound motion, the lines of 

 motion being all parallel and the rate of propagation being ica, to 

 the first order of small quantities the velocity, by a known theo- 

 rem, is equal to the product of tea and the condensation. Thus 

 the latter factor of the condensation is greater than the other in 

 the proportion of tc 2 to 1, and consequently for a given velocity 

 the condensation in the compound motion is in the same pro- 

 portion less than the condensation in the simple motion. 



If in the present instance of plane-undulations Vj be the com- 



