of a Liquid Mass without Weight. 367 



which seems necessarily to imply either a minimum or maximum 

 of extent ; and since it is evident that, with a given volume the 

 surface can always be increased by a proper change of form, it 

 has been concluded that a minimum ought to be chosen. Now 

 there was an equally legitimate intermediate supposition, which 

 has not been made, and which corresponds with the actual fact ; 

 that is, that, beyond definite limits, the surface is a minimum re- 

 latively to certain kinds of small changes of form, while it is a 

 maximum relatively to others. 



I prove the truth of this last principle by the study of the cy- 

 linder. Let us consider a liquid cylinder, terminated by two 

 solid bases, of any given length as compared with its diameter. 

 It is obvious, in the first place, that it may be made to undergo 

 small modifications of form which, without changing its volume, 

 increase its surface. This would evidently be the case, for in- 

 stance, if it were to become grooved with fine longitudinal ridges 

 and hollows, such that the sum of the ridges was equal in volume 

 to the sum of the hollows, each being measured relatively to the 

 original surface; and it is probable that it would also be the case 

 for any other modification which should change the figure of re- 

 volution. 



But let us suppose that the figure, without ceasing to be a 

 figure of revolution, were to transform itself into a succession of 

 alternately expanded and contracted portions, this change of 

 shape being of finite but excessively small amount. Then, if we 

 suppose the meridian line of the figure thus modified to be a curve 

 of sines, the area of the surface of the portion composed of one 

 expansion and one contraction can be found by calculation, the 

 condition that the volume shall not change being always kept in 

 view, and it is thus found that if the length of the portion in 

 question exceeds the circumference of the original cylinder, the 

 surface is less than that of the corresponding portion of the cy- 

 linder; and since the same result is applicable to each similar 

 portion of the entire figure, it follows that, under these circum- 

 stances, the total surface diminishes. The surface of the cylinder 

 is accordingly a maximum relatively to the kind of alteration we 

 have just indicated, and it is by a change of this kind that, as we 

 know, the spontaneous transformation of the cylinder takes place. 



Thus a liquid cylinder whose length exceeds its circumference, 

 or, in other words, in which the ratio of the length to the dia- 

 meter exceeds the quantity 7r, is necessarily unstable, because 

 the constant tendency of its superficial layer to diminish its area 

 favours the kind of change of shape which we have just been 

 considering. I may notice in passing that the theory of the 

 constitution of liquid veins which I have set forth at the end of 

 my Second Series depends upon this necessity. 



