146 Prof. H. Hennessy. On the Maximum [Dec. 20, 



where 9 is the arc bounding the segment filled with liquid. For this 

 segment u = Jr(l - V and therefore 



Q_ _ 



TU ~"~ c\ / 



ra 



5 is a small fraction compared to r and a, and if this expression is 

 developed we shall have very approximately* 



where K is a constant. This gives 



l dQ _ 3 (0 sing)*(l cosfl) (0-sin0)t 



26>! 



If we make ^ = 0, we shall have therefore 

 d0 



0= 



3 cos 0-2 



This equation may be satisfied by 9 0, o-r = f ^+7, a value less 

 than 27T. The first gives a minimum, the second a maximum. 



With 7 = 38 9' 56", |TT 4- 7 = 5-37850, 



3 sin = 1-85381, cos __ 5 . 37813 



^ 3 sin 6^ 



With 7 = 38 9' 57", |7r + 7= 5-37851, 



3 sin e = 1-85382, __ = 5-37848. 

 2 Sain 



With 7 = 38 9' 58", fw+7 = 5-37851, 



3 sin e = 1-85583, ' __ = 5*37882. 

 2 3sin6> 



With the first value the difference is +0*00037 ; with the third the 

 difference is .0-00031 ; consequently the value between both may be 

 considered as the nearest to the truth, and in this value the difference 

 is only 0*00003, or less than one-tenth of either of the others. If 7 = 

 38 9' [57", 6 = 308 9' 57' 7 , or a circular pipe, under the conditions 

 above mentioned, carries more liquid when filled up to this arc than 

 when quite full. 



* With the formulae of Chezy and Eytelwein, this would immediately follow. 



