VIII] OF AREAE MINIMAE , 569 



wall would be neither more nor less than the area a^. The calculation 

 is easy. 



The surface-area of a cylinder of length a is 27rr . a, and that 

 of our quarter-cylinder is, therefore, a.7Trl2; and this being, by 

 hypothesis, = a^, we have a = 7rr/2, or r = 2a/7r. 



The volmne of a cyhnder of length a is anr^, and that of our 

 quarter-cyhnder is a . Trr^/i, which (by substituting the value of r) 

 is equal to a^JTi^ 



Now precisely this same volume is, obviously, shut off by a 

 transverse partition of area a^ if the third side of the rectangular 

 space be equal to a/n; and this fraction, 

 if we take a = 1, is equal to 0-318... , or 

 rather less than one- third. And, as we 

 have just seen, the radius, or side, of 

 the corresponding quarter-cylinder will 

 be twice that fraction, or equal to 0-636 

 times the side of the cubical cell. 



If then, in the process of division of 

 a cubical cell, it so divide that the two 

 portions be not equal in volume but 

 that one portion be anything less than 

 about three-tenths of the whole or three- 

 sevenths of the other portion, there will be a tendency for the cell 

 to divide, not by means of a plane transverse partition, but by means 

 of a curved, cylindrical wall cutting off one corner of the original 

 cell; and the part so cut off will be one- quarter of a cylinder. 



By a similar calculation we can shew that a spherical wall, cutting 

 oiF one soHd angle of the cube and constituting an octant of a sphere, 

 would likewise be of less area than a plane partition as soon as the 

 volume to be enclosed was not greater than about one- quarter of 

 the original cell*. But while both the cyhndrical wall and the 



Fig. 220. 



* The principle is well illustrated in an experiment of Sir David Brewster's 

 {Trans. R.S.E. xxv, p. Ill, 1869). A soap-film is drawn over the rim of a wine- 

 glass, and then covered by a watch-glass. The film is inclined or shaken till it 

 becomes attached to the glass covering, and it then immediately changes place, 

 leaving its transverse position to take up that of a spherical segment extending 

 from one side of the wine-glass to its cover, and so enclosing the same volume of 

 air as formerly but with a great economy of surface, precisely as in the case of our 

 spherical partition cutting oflF one corner of a cube. 



