288 Becker — Geometrical Form of Volcanic Cones 



corresponding to this equation will differ only in scale or will 



2k 

 be geometrically similar. The value of — can easily be ex- 



P 

 pressed in terms of x and y. It will be found that 



4k 2 „ a 4k 2 



which gives 



2n_ y 



p (tan 2 5-l)-| 



# being the angle which the tangent at any point makes with 

 the x axis.* 



In the figure of least variable resistance the radius becomes 

 zero when $=45°. Below the point of the cone ■& increases 

 with the radius. In comparing the theoretical form with 

 actual occurrences this angle is especially significant. 



Comparison loith actual cases. — Whether or not the figure of 

 least variable resistance is that of a volcanic cone, can, I take 

 it to be, determined only by comparison in spite of the appar- 

 ently good reasons which have been stated for such a supposi- 

 tion. The first step for such a comparison is the reduction of 

 each drawing or of corresponding numerical data to the same 



2 k 

 unit, which can be done by help of the formula for — just 



given. Professor Milne gives diagrams and tabulated meas- 

 urements of Fusiyama and Kumagatake which I have 

 attempted to reduce in this way. The diagrams were taken 

 from selected photographs and are probably slightly but cer- 

 tainly not greatly distorted. Their actual scale is unfor- 

 tunately not given. For the left side of Fusiyama Professor 

 Milne gives in centimeters the position of eleven points on the 

 section referred to the axis of the volcano. Of these I rejected 

 the two uppermost for reasons to be mentioned presently, and 



2k 

 calculated — for each of the others, assuming that the cord 



connecting any two points was parallel to the tangent at a 

 point half way between them. By reference to Professor 

 Milne's diagram it will be seen that his points are so close that 

 * For the infinite logarithmic column on the other hand 



while for the catenary 



— — v 

 p tan# 



c / — x/c , jc/c\ 

 <= 2\ ) 



— — — - — — — =« cos ■&. 

 (tan 2 #+1) -J- y 



