462 Prof. Challis on the Fundamental Ideas of 



of the same quantities. Hence for a given value of r, the velo- 

 city w parallel to the axis and the condensation a are functions 

 of z—icat. We may therefore conclude that the composite velo- 

 city and condensation relative to an axis, so far as their expres- 

 sions to terms of the second order indicate, are propagated in 

 directions parallel to the axis with the uniform velocity ica. 



In the same article the reasoning is extended to the case of 

 vibrations relative to any number of axes having any positions in 

 space, terms of the second order being still included; and on 

 examining the conclusion of the investigation, it will be seen 

 that, for the case in which the axes are all parallel, the resultant 

 velocity parallel to the axes and the resultant condensation are at 

 each point functions of z — icat. Hence this composite velocity 

 and condensation relative to any number of parallel axes, so far 

 as terms to the second order indicate, are propagated with the 

 uniform velocity ica. It has already been argued, when terms 

 of the first order were alone considered, that if there were an un- 

 limited number of sets of vibrations relative to parallel axes, and 

 all in the same phase of vibration, the transverse vibrations 

 would be neutralized, and the resultant motion be in parallel 

 straight lines. And clearly this condition can be fulfilled when 

 the motion and condensation expressed by terms of the second 

 order are included, these being much smaller than the motion 

 and condensation expressed by those of the first order. Now, 

 when the motion is in parallel straight lines, and the velocity 

 and condensation are propagated with a uniform velocity, it may 

 be proved (see Phil. Mag. for September 1865, p. 218) that 

 there exists between the velocity V, the condensation S, and the 

 rate of propagation xa, the relation expressed by the equation 



V V 2 



Ka K*a z 



For composite vibratory motion, supposing the transverse mo- 

 tions to be neutralized, the general value of V to the first order 

 of small quantities is 2. [m sin q(z— icat + c)'], the number of 

 terms being unlimited, and the coefficient m being the same for 

 all and as small as we please. Hence, grouping the terms ac- 

 cording to the values of q, and supposing n to be the number of 

 terms in any one group, it may be shown, as in the May Num- 

 ber (p. 349), that the composite velocity for that group is 



mrr sin q(z — fcat + 6). Hence the whole composite velocity is 



mS . [n 2 sin q [z — icat -\-6)~], 



n, q, and being in general different for each term embraced 

 by 2. In order to obtain S to the second order of small quan- 



