218 Prof. Challis on Hydrodynamics. ■ 



such axes, both which, as well as the equation itself, are inde- 

 pendent of arbitrary circumstances of the motion. From the 

 first law we may infer that dr=dr ! , the time being given. The 

 other, after supposing the motion to be composite in the manner 

 stated in art. 34, gives rise to a certain relation between V and 

 p, which is investigated in an article " On some Hydrodynamical 

 Theorems," in the Philosophical Magazine for November 1853, 

 and also less completely in the October Number of 1862 (art. 

 35). The latter investigation only takes account of changes of 

 velocity accompanied by changes of density, while the other in- 

 cludes changes of velocity due to divergencies of the lines of 

 motion, and on that account is proper for our present purpose. 

 The reasoning is briefly as follows. Conceive a slender four- 

 sided tube to be formed by four planes, two passing through one 

 of the focal lines and two through the other ; and at the time 

 t let V and p be the velocity and density of the fluid, and m the 

 transverse section of the tube, at the distance z measured along 

 its axis from an arbitrary origin ; and let at the same time V, p' t 

 m' be corresponding quantities at the distance z + Sz. Then the 

 increment of fluid in the space between the sections in the small 

 time St is ultimately 



(Ypm-Y'p'm'jSt. 



Also the difference between the condensations in two adjacent 

 spaces, each of length 8z, terminated towards the same parts by 

 the sections m and m r } is ultimately 



(p — 1 ) mSz -— (p ! — 1 ) m'Bz . 

 If the former of these quantities be equated to the latter, St will 

 be the time in which the condensation is propagated through Sz. 



Sz . 

 Hence ~- is the rate of propagation ; and as this, by the argu- 

 ment above, is equal to ica, we have, after passing from differ- 

 ences to differentials, 



d.Ypm_ d.{p—l)m 

 az dz 



Hence, by integration, 



Vp=/az(p-i) + M). 



m 

 Since m varies as the product rr', this equation may be changed to 



Eliminating Yp from (e) by this formula, the result is 



