328 Mr. Lubbock on Cask-Gaging. 



which BO is the semiaxis minor, the axis major coinciding in 

 direction with Ox. 



In the second variety the arc ABC is considered to be a 

 portion of a parabola whose vertex is B, and whose axis co- 

 incides in direction with BO. 



In the third variety the arc BC is considered to be a por- 

 tion of a parabola whose vertex is in the line Ox beyond D. 



In the fourth variety BC is a straight line. 



In works on gaging, the second and third varieties are sup- 

 posed to be intermediate between the first and fourth, as in the 

 figure, which is copied from Symons's Practical Gager, p. 187; 

 but this is incorrect, for the second variety, as thus defined, 

 coincides ver}- nearly with the first, and the third with the 

 fourth. 



Let BF (the internal bung diameter) be called b. 

 CE (the internal head diameter) be called h. 

 HD (the internal length) be called I. 



In the first variety the equation to the curves ABC, GFE, 

 in the plane xz, is 



z - = ^b'-~ w -x: 



In the second variety the equation to the curve ABC is 

 07 b-h 9 



In the third variety the equation to the curves BC, FE, is 



z- = 4& 3 - 



2(b°--h*) 



I 

 In the fourth variety the equation to the curve BC is 



z = 2b j-x. 



In the first variety the content of the cask is easily found 

 by means of the triple integral fffdx, dy, &z taken between 

 the proper limits, 



=^J-i~2& 3 + # ! \ (tt = 3-14159). 



In the second variety the content 



20 LS 3 J 



In the third variety, 



=!-*{*•+ *}. 



In the fourth variety, 



