21-23] STEADY MOTION. 23 



with two similar equations. Multiplying these in order by 

 dx/ds, dy/ds, dzjds, and adding, we have 



du dv dw _ cft __ 1 dp 

 ds ds ds ds p ds 9 



or, integrating along the stream-line, 



P 



This is similar in form to (2), but is more general in that it 

 does not assume the existence of a velocity-potential. It must 

 however be carefully noticed that the constant of equation (2) and 

 the C of equation (3) have very different meanings, the former 

 being an absolute constant, while the latter is constant along any 

 particular stream-line, but may vary as we pass from one stream 

 line to another. 



23. The theorem (3) stands in close relation to the principle 

 of energy. If this be assumed independently, the formula may be 

 deduced as follows*. Taking first the particular case of a liquid, 

 let us consider the portion of an infinitely narrow tube, whose 

 boundary follows the stream-lines, included between two cross 

 sections A and B, the direction of motion being from A to B. Let 

 p be the pressure, q the velocity, ft the potential of the external 

 forces, a the area of the cross section, at A, and let the values 

 of the same quantities at B be distinguished by accents. In each 

 unit of time a mass pqa at A enters the portion of the tube 

 considered, whilst an equal mass pq a leaves it at B. Hence 

 qa = q a. Again, the work done on the mass entering at A is 

 pqa per unit time, whilst the loss of work at B is p qa . The 

 former mass brings with it the energy pqa (^ q 2 + ft), whilst the 

 latter carries off energy to the amount pqo- (^q 2 -f ft ). The 

 motion being steady, the portion of the tube considered neither 

 gains nor loses energy on the whole, so that 



pqo- + pqo- (%q 2 + ft) =p q a + pq a (%q z + ft ). 

 Dividing by pqa (= pq a }, we have 



* This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica, 

 Argeiitorati, 1738. 



