UTanchesfer Memoirs, Vol. xliv. (1900), No. 10. 



We therefore have 



du 

 f{u + ii>)f'{i/-il>)j^ = c, 



jf'{u + /7>)f\it - ii') du = d + , 



(2). 



Here a and l> of the Lai^raiic^ian equations are introduced 

 as constants of integration. The analogy with the other 

 method is evident, b corresponding with ^. 



Turning to the equations to find the pressure, we have 



d \ p , .„ ,„, ) d I ■ dx ■ dv\ 



d 



In our case 



etc. 



dv ., dy du 



X — + J — = r — , 



da da da 



■ dx ■ dv du ., . 



X — H- y ^ = r — , SO that 

 db -^ db db ' 



^// + 



da 



1 1 d ( du\ 



with a precisely similar equation. 



The condition that the motion shall be irrotational 

 requires that c shall be independent of a and b, which is 

 satisfied in the case of steady motion when c is an absolute 

 constant. The pressure is then given by 



fi 1 



- + gy + ^^^'= constant, 



P 



t c" 



or c + £j; + _ If (^1/ 4. //;) f i^u - ib) = constant . . (3). 

 P 2 



Comparing this with 



^~=g{C-y)--^ 



P ' 2f{<p + i^)/'{<t>-i^P) 



used in the Eulerian notation, we see that the two forms 



