12 Prof". A. Gray: Notes on Hydrodynamics. 



The projections of AB on the axes after dt has elapsed are 

 therefore 



dx ( 1 + s— dt ), ^— ■ dx dt, - 7 — dx dt ; 

 \ ox J o% dx 



and if small quantities of the second order be excluded the 

 new length of AB is 



ou 



*(!+£*} 



Similarly we obtain for CD in its new position projections 

 on the axes which can be written down from those for AB, 

 and the length of CD has changed from dy to 



<r(i+g*> 



Further, it can be shown that to the degree of approxi- 

 mation specified the angle </> between the new lines AB, CD 

 (which intersect in the new position of P) is given by 



cos 



mi+rK 



It follows that the former area dx dy has in its new position 

 the value 



^{i+(g4;>} 



and the unital time-rate of areal expansion is 



~du ~dv 

 O^x ~dy 



following the fluid, that is the statement made in § 12 is 

 justified. 



Some further notes regarding vortex-motion, dealing 

 principally with surface and volume integrals, are reserved 

 for another communication. 



