Prof. Challis on Hydrodynamics. 35 



is obtained, for the value of the condensation (o 1 ) along the axis, 

 , km . 2mV (tc 2 — l)m 2 . 2 



^I 8m ^ 8flV-l) COg¥+ ~~^ sin-^ + &c. 



Now, since udx + vdy + wdz is strictly an exact differential only 

 for indefinitely small distances from the axis, the above value of 

 / is strictly true only for indefinitely small values of r ; so that 

 for the purposes of exact calculation /= 1 — er 2 . On the con- 

 trary, the exact values of w' and o 1 are expressed by series con- 

 sisting of an unlimited number of terms, each of which is a 

 circular function of z-j~a Y t-\-c. These expressions accordingly 

 prove that w' and o' are propagated along the axis with the 

 same constant velocity a v and without undergoing alteration. 

 The proof of this law is independent of the value of m; and 

 whether w' and cr l be large or small, the above series retain the 

 same form, the terms following the first or principal terms 

 always coexisting with the latter. For our present purpose it 

 suffices to include only terms of the second order with respect 

 to m, in which case k is a numerical constant, the analytical ex- 



pression for which is ( 1 H ^ ) • 



Again, it is to be remarked that since explicit formulae ex- 

 pressing the velocity and condensation in vibratory motion have 

 been obtained without reference to a given mode of disturbance, 

 in every case of disturbance producing vibrations of the fluid the 

 motion must in a certain manner be compounded of the motions 

 defined by these formulae. It is therefore requisite to inquire, 

 next, respecting the laws and the effects of this composition. 



From the usual approximate equations 



a 2 .do- du n a 2 . da ($v a 2 . do- dw 



dx T dt ' dy ^ dt ' dz T dt 



the known theorem that udx + vdy + wdz is approximately an 

 exact differential for small vibratory motions may be deduced 

 prior to the consideration of any arbitrary disturbance. Let us 

 suppose that in this case also that differential is equal to (d ./</)), 

 / and cf) being functions of the same variables as before. Then 

 from the above equations, and from the approximate equation of 

 constancy of mass, viz. 



do du dv dw 

 llt + dx + dy + ~dz~ =0 > 

 there results, 



9 W + dy 3 / ^ J dz 2 a? df -U - 



The assumed compositions of /and <j) are satisfied by resolving 



J)2 



