Prof. Challis on Hydrodynamics. 259 



the condensation (a') and velocity (w 1 ) along a rectilinear axis of 

 motion : — 



aa' = niK sin ty ~ 3a ^ 2 _^ cos 2bfi + — (* 2 - l)sin 2 V, («) 



w' = msinV-3^7^jyCos2V, (£) 



/u, being put for z— Kat + c, and b for -^-, and the other letters 



A 



having their usual significations. The first terms of these ex- 

 pressions are much more considerable than the others, on account 



m 

 of the factor - contained in the latter, which is supposed to be 



a very small quantity. It was argued (p. 212) that the last 

 term of the value of aa f necessarily coexists with the first, other- 

 wise the movement of a given particle could not be wholly vibra- 

 tory. Further, it was argued that as the equations (a) and (/3) 

 were obtained without reference to any specific case of motion, 

 they must be applicable to all cases of vibratory motion; and 

 that since the antecedent reasoning showed that motion along 

 an axis is always accompanied by motion transverse to the axis, 

 such general application requires that the total motion in a given 

 instance should be composed of an unlimited number of longi- 

 tudinal motions, such and so disposed that the transverse mo- 

 tions destroy each other. This being understood, we may at 

 present omit the consideration of the terms involving m 2 , and 

 confine ourselves to quantities of the first order. In that case 

 the differential equations from which the values of a 1 and w' are 

 derived are linear with constant coefficients, and we may conse- 

 quently assume that any arbitrarily imposed velocity is the sum 

 of an unlimited number of velocities defined by the expression 



* 2tt 

 msin— - {z— /cat-\-c), m, A, and c being at disposal, and the 

 A 



axes of motion being all parallel ; and similarly that any con- 

 densation resulting from arbitrary conditions is compounded of 

 an unlimited number of condensations accompanying the veloci- 

 ties and severally defined by circular functions. But if V L be 

 the composite velocity and a l the corresponding composite con- 

 densation, the relation between V x and <r l will not be the same 

 as that between a' and w' in each component, although the 

 rate of propagation of velocity and condensation will be the 

 same in the compound waves as in the components, namely, the 

 velocity /ca. The reason for the different relation will be seen 

 by considering that in an uncompounded series of vibrations the 

 condensation is due both to longitudinal and to transverse vibra- 



S2 



