Mass carried forward by a Vortex. 611 



•circle k— sin 75°. Here 



log X= 1-61752, 

 logp =1-66316, 

 log A= 1-77840 



= logsinh(4'78 + -._-)= log sin h 4-7881, 



log sinh 1-5960= '37388 

 add 1-38315 



io «V(^ 



log a 2 = 



1-75703 ==log -571,5.2 



\pgpfa = 



1-66316 





-1-87851 





T78465 = log -60905 



9 



p a~ 

 2~a ~ 4 = _ 



•30452 



- -14288 





•16164 



o a 2 \ 

 r a ~~l)~ 



1-87851 

 1-60427 



1-48278 = log -30394 

 : -30452 + -30394 = -60846. 



a 



The point on the boundary is then found either graphically 

 by the point on the circle k = sm 75° whose distance from the 

 axis is '608, or it may be calculated direct as y = '115a. 



The area of the loop measured graphically from the curve 

 is "1099a 2 and the distance of its centre of gravity from the 

 axis ='8112 a. The volume of the liquid carried forward is 

 therefore '5600a 3 = 1181 m. 



The energies. — The respective energies are now de- 

 termined by substituting the values of ^ 2 , ^, and the 

 graphically measured volumes in the expressions already 

 determined. The value of ^ 2 is obtained by inserting the 

 co-ordinates of the point where the boundary of the core 

 cuts the equatorial plane, viz. 



,i> — a — b, k' = b/(2a — b), 



where 



F=log^ + i^(L-l)... 



E=i + } 2 r\L-\)... 



