330 Mr. Lubbock on Cask-Gagvig. 



sidering the cask, with the same length, bung and head diame- 

 ter, is thus in this instance 6'46 gallons, or about 5'5 per cent. 



The term ullage appears to be used indifferently to denote 

 either the liquor contained in a cask, or the vacuity above the 

 liquor. It has been seen that accurate, that is to say, rigorous 

 and extremely simple expressions can be obtained for the en- 

 tire content of a cask of either variety ; but it seems to me im- 

 possible to find expressions sufficiently simple to be used in 

 practice for the ullage of a cask lying upon its bulge, except iu 

 the case of the first variety, and when the dry inches do not 

 exceed the half difference of the bung and head diameters. 

 This difficulty arises from the limits of the integral which has 

 to be employed. 



The cask considered as of the first variety is a portion of 

 the spheroid, of which the equation is 



p ( s « + f) + (b 3 -h 2 ) X 2 = 4,1)* P. 



If this spheroid be intersected by a horizontal plane, the 

 curve of intersection will be an ellipse, of which the semiaxes 



ate l*/~W^P , 



and v/'4i 2 — s 2 : 



</ b*-h 2 



the area of this ellipse = — ^ — ' 



This quantity multiplied by ds and integrated between the 

 proper limits will give the ullage in the first variety ; and I 

 find that if d represent the dry inches, or the distance from 

 the bung-hole at B to the surface of the liquor, 



the ullage or vacuity = (— b— d), 



when the dry inches do not exceed the half-difference of the 

 bung and head diameters, that is, when the surface of the li- 

 quor is not beneath a straight line joining A and C. I under- 

 stand that in practice this condition generally obtains, except 

 where a cask has leaked, or where the liquor has made a long 

 voyage, as in the case of rums. This formula might be use- 

 ful if casks could be considered as belonging to this variety, 

 but of course the difference of figure has a sensible effect 

 upon the ullage. 



In the other varieties, and in the first variety except in the 

 particular case I have mentioned, the integral which has to 

 be taken in order to obtain the ullage is extremely compli- 

 cated ; and I apprehend the simplest way to arrive at the ul- 

 lage in all these cases is to construct Tables for various casks 

 by quadratures, that is, to divide the liquor into parallel and 

 thin slices, vertical and perpendicular to the axis of revolu- 

 tion, each of which may be considered as a small cylinder of 

 which the base is a circular segment. 



