21-23] STEADY MOTION. 23 



with two similar equations. Multiplying these in order by 

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



du dv dw dl 1 dp 



u^r+v^r+w^r^ ~j --- ^ 

 ds as ds ds p ds 



or, integrating along the stream-line, 



S = -n-Jg. + C ........................ (3). 



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, H 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 

 qo- = q'o-'. 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 pq<r(&f + &)t whilst the 

 latter carries off energy to the amount pgV (i^* + &') The 

 motion being steady, the portion of the tube considered neither 

 gains nor loses energy on the whole, so that 



pqa + pqa (\q> + fl) =p'q'a f + pq'a' (%q"> + Q'). 

 Dividing by pqa (= pqa), we have 



+t fl 4n-12L + ij*+fl / > 



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

 Argentorati, 1738. 



