2729.] STREAM-LINES. 23 



with two similar equations. Multiplying these in order by 



dx dy dz ,,. 



j-, -f-. -T-, and adding, we have 



as as as 



du dv dw dV 1 dp 



u -T- + v -j- + w --T- = - -j f- , 



as as as as p as 



or, integrating along the stream-line, 



-V- q *+C (9). 



P 



This is of the same form as (8), but is more general in that it 

 does not involve the assumption of the existence of a velocity- 

 potential. It must however be carefully noticed that the constant 

 of equation (8) and the C of equation (9) 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. 



29. The formula (9) may be deduced from the principle of 

 energy without employing the equations of motion at all. Taking 

 first the particular case of a liquid, let us consider the portion of 

 an infinitely narrow tube, whose walls are formed of 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, F 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 pqcr 

 at A enters the portion of the tube considered, whilst an equal 

 mass pqa leaves it at B. Hence qa = qa. Again, the work 

 done on the mass entering at A is pqa per unit time, whilst the 

 loss of work at B is pqa . The former mass brings with it the 

 energy pqa ( J &amp;lt; + F), whilst the latter carries off energy to the 

 amount pqa (J q z + F ). The motion being steady, the portion of 

 the tube considered neither gains nor loses energy on the whole, 

 so that 



pqa + pqa (\q* + F) =p qo + pqa (?q 2 + F ). 

 Dividing by pqa (= pqa), we have 



