■".■"•.< ■■■■•■, ';..v\r. 

 Prof. Challis on the Principles of Hydrodynamics. 37 



'uu h-y/s. 



spherical condensed wave, the breadth of which is -, will be pro- 



pagated from the centre with the velocity a, and according to 

 the above formula the condensations at correspondmg points of 

 the wave at different distances, will vary inversely as the distances. 

 But it is certain that under these circumstances the condensa- 

 tions must vary inversely as the square of the distances, in order 

 that, in conformity with Axiom I., the quantity of fluid may re- 

 main the same. Here, then, we have another contradiction indi- 

 . eating that a logical fault has been committed. 



It may be remarked that, in the above argument, the reason 

 for the discontinuity of the function / is drawn from a funda- 

 mental property of the fluid, and not, as is usually done, from 

 independent analytical considerations. The legitimacy of this 

 method cannot be questioned. I confess that I am imable to 

 comprehend how, on any other principle, it can be shown that 



— may vary per saltum, and that cr must not vaiy per saltum. 



Example III. Let the fluid be incompressible, and let the 

 motion be jjarallel to a fixed plane. 



Supposing that iv = 0, the equation (2.) becomes for this case, 



du ^^ _o 



dos dy 

 For the purpose of obtaining an integrabfefeffuation, it will be 

 assumed that udx + vdy is an exact differential {d9). Thus the 

 above equation becomes 



dW_ . d^ _^ 



dx" "^ drf ' 

 the integral of which is 



Hence 



dx 



..Y>{x+y V -\)+f{x-y V -\), 



and 



JQ 



To take a particular instance, let 



\<{x + y ^/ -\)= ^ [x + y ^/ -\), 

 and 



Then n = mx, v= —my, and the equation (1.) becomes by in- 



