and the Elastic Limit of Lava. 285 



column is in a condition of stabLe internal equilibrium, the 

 total resistance of any section to crushing stress must at least 

 equal the .pressure of the load resting upon the same section, 

 so that if p is the pressure of a load which would strain a given 

 section to its elastic limit, or the possible load, and if I is the 



actual load, p = l and -57= 1- The square of this differential 



coefficient, if not unity, must also exceed it, so that 



dl 2 + ' 



where F denotes some positive quantity the limiting value of 

 which is zero. It follows from the definitions that 



p=7tKy* ; l=7tpfy 2 dx; 



X 



and therefore 



ay_/2uV y^ 



^ 2_ W 7 

 the accent as usual denoting the differential coefficient taken 

 with regard to x. The volume of the solid of revolution above 

 the section at x is therefore 



n ly 2 dx=7t I ( — 1 y' 2 dx—7t J Fy 2 dx. 



XX X 



For the limiting value F=0, or when the load everywhere 

 exactly equals the carrying power, these equations give for the 

 generating curve 



y=Ae- xp / 2K , 



a well known formula for this condition. Since this logarith- 

 mic curve meets the axis only at infinity, the condition l=p 

 answers only to an infinite value of a. 



In the equation for the volume, either term may be regarded 

 as varying independently. If the term containing y' is sup- 

 posed constant, the volume above any and every section will 

 be a maximum, for stated values of the constants, when the 

 term containing F is a minimum ; and this must be true even 

 when a is not given the special value oo. Consequently when this 

 term is a minimum, each section of the column will be loaded 

 as nearly to the elastic limit as the conditions will permit and 

 the entire column will contain less material than any other 

 stable column of the same height, or, for a given amount of 

 material, will be the highest stable column which can be 

 generated by the revolution of a continuous curve, or which 

 can be formed under the action of a uniform law. This postu- 

 late of continuity of course excludes from the possible finite 



