2 PROFESSOR ANDREW GRAY, 



the velocity in the direction of ds' at a point Q, at a distance ds in advance of P. 

 along the line of flow, is q' + dq'/ds . ds. After the lapse of the interval dt, the 

 particle which was at P has reached Q, and its velocity (along ds') has become, 

 to the first order of small quantities, q' + dq'/dt . dt + dq'jds . ds. Since q = ds/dt, 

 the particle when at P is thus gaining velocity in the direction of ds' at rate 



dq dq dq' 



0* = ¥ + «W .... (i). 



3. An elementary point of some difficulty to the student arises in this connection. 

 The integral 



J dt 

 is to be differentiated with respect to ds', specified as in § 2. The result is 



*&*-%*-% .... (2). 



ds J dt dt ds dt 



Here we have ds/ds' = cos 0, where 6 is the angle between ds' and the terminal element 

 of the line of integration. It is at first a little difficult to see that ds/ds' has this 

 value. But the whole question is, What is the change in fdq/dt . ds produced by 

 a small step ds' inclined at an angle 6 to ds ? Now the step ds' may be regarded 

 as made up of a step ds' cos along the stream-line, followed by one ds' sin 6 

 at right angles to the stream -line. The former gives the element dq /dt . ds' cos 6 of 

 the integral ; the latter, being at right angles to the stream-line, leaves the integral 

 unchanged ; therefore the result of the differentiation is dq/dt . cos 0, or dq'/dt. 



The student is at first tempted to take ds' as one component of a step ds along 

 the stream-line, that is, as ds cos 6. He forgets that if this is done the element 

 dq/dt . ds of the integral, which he now makes to correspond to ds', is the result, not 

 of the step ds' alone, but in part also of the coexistent step ds sin 6. 



4. The flow of energy in the hydrodynamic field may be discussed in two ways. 

 We may determine the rate of change of energy within a closed surface S fixed 

 in the fluid, or we may find the rate of change of the energy of the definite mass 

 of fluid which at time t occupies the space within the given surface S. These are 

 two distinct problems — since at time t + dt the fluid which at time t occupied the 

 space within the surface no longer does so precisely — and a comparison of their 

 solutions is instructive. As we shall see presently, the solution of one problem 

 can be derived from that of the other ; and the application of the results to 

 particular cases gives various well-known hydrodynamical theorems. 



5. We take first the problem of the rate of change of the energy of the mass 

 of fluid contained at time t within the surface S. Let q denote, as usual, the 

 resultant velocity of a particle of the fluid, and q its total time-rate of variation. 

 If, as is usually the case, the external forces (per unit mass) be derivable from a 

 potential — which we shall call V — we have 



3V q dp 

 qq = -q — - J- £-. 

 HH 'ds p ds 



