354- Prof. Challis's Determination of the Velocity of Sound 



ods , du ^ ^ds ^ dv .ds dw __ 



dx dt ' dy dt dz dt 



ds du dv , ——o 

 di'^di'^'dj'^Tz' 



all the differential coefficients being partial. Hence by inte- 

 gration, 



p ^ds ,, , d.fahdt ,^ 



u— —I a^ — dt + c =— ♦^ +c 



»/ ax fij^ 



r.= -ra-%dt^c^=-^jf^^d 

 -^ dy dy 



r^ds ,, d.faHdt ,u 



J dz d^ 



the arbitrary quantities c, c', c" being functions of ^,3/ and «. As 

 the motion is by supposition vibratory, it will be assumed that 

 c=0, c' = 0, c" = 0, which is to assume that no part of the velo- 

 city is independent of the time. Now substitute vf/ for —Jahdt. 



Then 



d-h t/4/ f/v^ 



rf.r Gj/ dz 



It follows that udx + vdy + wdz is an exact differential. 



Here it must be particularly remarked that the above result 

 has been arrived at prior to the consideration of any particular 

 case of disturbance. Consequently ndx + vdy + wdz is an exact 

 differential for some reason which applies equally to all cases 

 of small vibrations. Such a reason would be given if it were 

 proved, that the motion in every case is composed of motions in 

 plane-'waves, that is, waves in which the motion of each particle 

 is perpendicular to a fixed plane and a function of the distance 

 from the plane, the number of such waves and the directions 

 of motion being taken arbitrarily. The consequences of the 

 general supposition of plane-waves have been traced in an in- 

 genious paper by Mr. Earnshaw, contained in the Transac- 

 tions of the Cambridge Philosophical Society (vol. vi. part 2, 

 p. 203). It is there shown that the velocity of propagation on 

 that supposition is the constant «, which, as will afterwards 

 appear, is a first approximation to the true theoretical velocity 

 of sound. Mr. Earnshaw finds also that "a plane-wave can- 

 not be transmitted through any fluid unless it extend com- 

 pletely across the medium from boundary to boundary." This 

 result makes it impossible to conceive how the motion can in 

 every case be composed of motions in plane-waves. There is, 



