284 Becker — Geometrical Form of Volcanic Cones 



analogous to Professor Milne's but differ in some important 

 particulars, and the data accumulated by him enable me to 

 carry the subject a little further. Professor Milne regards the 

 problem of a volcanic cone as that of the form which would be 

 assumed by a mass of loose ash, and he is led to the conclusion 

 that the form which would be assumed by such material 

 would be that generated by the revolution of a logarithmic 

 curve round its asymptote.* His experiments on sand, how- 

 ever, as he points out, gave him only right cones with straight 

 sides, excepting when material of different sizes was employed 

 and the action was such as to involve a sorting of the particles. 

 I cannot think that the form generated by the revolution of a 

 logarithmic curve about its asymptote is the figure of stability 

 for a mass of loose material. Without any analysis it seems 

 beyond question that were a long trumpet-shaped tube filled 

 with dry sand, inverted on a horizontal plane and the tube 

 withdrawn, the column of sand would collapse, and that a cone 

 or some figure very similar to it would result. On the other 

 hand it is a well known fact that the solid produced by the 

 revolution of the logarithmic curve about its asymptote is the 

 form which a loaded column of uniform strength must have 

 when the material of the column is continuous and coherent 

 like metal or stone. Such a column may be cut at any point 

 and the load it will bear is the weight which the infinite col- 

 umn above this section would possess were the material 

 tiniform.f This, however, is not exactly the problem of the 

 volcanic cone, which is an unloaded finite column, and neither 

 an infinite one nor, what amounts to the same thing, a finite 

 one supporting an extraneous weight. 



Solid unloaded column of " least variable resistance." — A part 

 of the ejecta of a volcano is melted lava and congeals to rock. 

 A large part is also in the form of sand or " volcanic ash," but 

 of this much is induiated to a tolerably firm material after 

 ejection by one cause or another and, although weathering and 

 frost will often cover the slopes of a mountain such as Fusi- 

 yama with detritus, it appears to me that volcanic cones must 

 be regarded as essentially continuous masses. 



Suppose a columnar mass of uniform coherent material, the 

 surface of which is generated by the revolution of some curve 

 about a vertical axis. Let the height of this column be a, its 

 radius y, the distance of any horizontal plane from the base x, 

 the specific gravity of the material p, and the coefficient of 

 resistance to crushing stress at the elastic limit x. If such a 



* Professor Milne does not give his conclusion in this form, but states that the 

 sum of equally spaced ordinates will be to their difference in a constant ratio, 

 which is equivalent to the statement in the text. 



f The volume of this solid of revolution is of course finite in spite of the fact 

 that its length is infinite. 



