Matter and Force in Theoretical Physics. 461 



Hence at points contiguous to the axis the transverse velocity 

 and condensation are expressed by formulae exactly analogous to 

 those which express the direct velocity and condensation, and, 

 the direct velocity of propagation being /ca, the transverse velo- 



city of propagation is K f a. Also, substituting -^ for e in the 



value of k, we have /c = ( 1 + — ^ J = ( 1 + -72 ) ; so that 

 /c' 2 = g -. But another relation between k! and tc may be ob- 

 tained by a consideration of the apparent elasticities in the two 

 directions. From the foregoing values of w and cr for a single 



series of waves, it follows that w= — . If the mption had been 



K 



in parallel lines, and therefore unaffected by transverse action, 

 the relation would have been w — ao. Hence the transverse 

 action diminishes the velocity corresponding to a given condensa- 

 tion in the ratio of - to 1, and consequently changes the actual 



K « 2 



elasticity a 2 into the apparent elasticity -%. Now this trans- 

 verse action may be regarded as the effect of a tendency of the 

 direct vibrations to lateral spreading ; in which case the direct 

 vibrations have on the transverse vibrations just the inverse effect 

 of the transverse on the direct, and consequently change the 

 actual elasticity a 2 in the transverse direction into the apparent 

 elasticity K 2 a 2 . The ratio of this apparent elasticity to the other 

 is k 4 ; and the same ratio, inferred from the squares of the velo- 



K n 1 



cities of propagation in the two directions, is —§, or -% — ^ . 



Hence tc 6 — /c 4 =l. The numerical value of k is thus found to 

 be 1-2106. 



The other point which I purposed to consider is the relation 

 between the composite velocity V and the composite condensa- 

 tion S when the motion is in parallel straight lines, and terms 

 of the second order are taken into account. In the % May Number 

 I have obtained, on the supposition that (dyjr) = udx + vdg + wdz, 

 a value of -^ to terms of the second order, applicable generally 

 to vibrations relative to a single axis, viz. 



^=-m2.pcos?£]-m 2 A2. £- sin2gf] +m 2 Q. 



As J was put for z — tcat + c, and /and g are functions of r the 

 distance from the axis, and as the equation which determines Q 

 proves it to be a function of z — /cat and r, it follows that the 

 complete value of yjr to terms of the second order is a function 



