Prof. A. Gray : Notes on Hydrodynamics. 3 



X is, in the case of steady motion, constant from stream-line 

 to stream-line. If there is no elemental rotation anywhere 



throughout any finite portion of 

 the fluid in steady motion the value 

 of x i s the same for all stream-lines 

 in that portion. 



4. To prove that co ss r, defined 

 as above, has the value given b} r 

 the equation 



~ds "?>" 



2o} S s' sin 6- 



-51 ( 7 ) 



we may proceed thus. Consider 

 the parallelogram (see figure) of 

 which adjacent sides are PP', 

 PQ, that is ds', ds. The average 

 velocities of the fluid along the 

 four sides PQ, QR, PP', P'P are 



13? 

 2 3* 



Q+ s^-ds, q'+^ds+- c 



3* 



13<?' 



2 3. 



-,ds' 



K* + ^ + tH-(^sH 



The first and third of these multiplied by ds, and the second 

 and fourth multiplied by ds', give a sum of products which 

 is the circulation round the parallelogram. The sum is 

 (dq'l?)s — ~dq/~ds') ds ds'. In other words, if q c be the com- 

 ponent of velocity along the boundary of the parallelogram 

 at any element dc, we have for the parallelogram 



Mf-» 



ds ds' '. 



But if P" be a point within the parallelogram in the 

 plane of ds and ds', and p be the perpendicular distance of 

 P" from the element of periphery dc, the angular velocity 

 of a fluid particle at dc about P" is (jc/p. Thus we have 



f 



P \ds 





ds di 



If (o ts > be the mean angular velocity about P" for the 

 particles of fluid on the periphery at the instant considered 

 we get, since J p dc is twice the area of the parallelogram, 

 and this is also 2 ds ds' sin 6, 



■bq 



3*'' 

 Thus (7) is proved. 



B2 



)„, sin 0=~- 

 ds 



