VIIl] 



OF AREAE MINIMAE 



353 



partition, AD, since BD = 1, will always equal tan^ (/3 being 

 the angle ABC). And the oblique partition, GH, being equal to 

 AB/\/'2=^ l/-V2co& ^. If then we call our vertical, transverse 



and oblique partitions, V, T, and 0, we have F = tan j8 ; 

 T = V2 ; and = 1/a/2 cos j8, or 



V -.T :0 = tan^/V2 : 1 : 1/2 cos ^. 



And, working out these equations for various values of ^, we 

 very soon see that the vertical partition (F) is the least of the 

 three until j8 = 45°, at which limit F and are each equal to 

 1/V2 = -707 ; and that again, when ^ = 60°, and T are each 

 = 1, after which T (whose value always = 1) is the shortest of 

 the three partitions. And, as we have seen, these results are at 

 once appHcable, not only to the case of the plane triangle, but 

 also to that of the conical cell. 



Fig. 141. 



In like manner, if we have a spheroidal body, less than 

 a hemisphere, such for instance as a low, watch-glass shaped 

 cell (Fig. 141, a), it is obvious that the smallest possible 

 partition by which we can divide it into two equal halves 



T. G. 23 



