of the Motion of Fluids, SfC. 389 



The differential coefficient -^ is the ratio of the increment of 



at 



velocity to the increment of time, considered independently of 

 the change of space ; the integral indicated above is to be per- 

 formed in reference to s, t being constant. The preceding reason- 

 ing shews that both the line s and the function -~ may be 



discontinuous. It is not easy to apply the equation (D), on 

 account of the difficulty of determining the values of a, li, and 7, 

 which fix the position of the line s, and of ascertaining the 

 velocity at every point of this line, in terms of the velocity at 

 the j>oint where the pressure is known. When, however this 

 has been effected, we may obtain, by means of the known pres- 

 sures at two points of this line, an equation proper for determining 

 the velocity at one of the points at any time, when the velocity 

 is given at a given time ; and by inference the velocity at every 

 point of the line. Thus the problem will be completely solved. 

 The process here indicated, is that which has in fact been 

 adopted in the problems which have admitted of solution; but 

 in most cases the mathematical difficulties are too great to be 

 overcome. I proceed to take one or two simple instances. 



(1). Conceive the disturbance to be made in any mass of 

 fluid, acted upon by no forces, by a spherical body expanding 

 or contracting according to a given law, and in the same degree 

 in all directions from the centre. 



Hence, p,- E^MX . •£ -f^,). 



