286 Becker — Geometrical Form of Volcanic Cones 



forms the truncated logarithmic column, with or without an 

 imposed load. It appears to me that a volcanic cone, formed 

 as indicated at the beginning of this paper must fall under this 

 problem. 



For the sake of brevity, I propose to call the solution of this 

 problem the form of least variable resistance, a term intended to 

 include as a special case the form of uniform resistance or the 

 infinite logarithmic column. The form of least variable resist- 

 ance then is a solution of the equation 



or by the method of variations 



giving 



or y=As ' + Be Vl , 



where A and B are arbitrary constants. 



If the origin of the curve is now taken at the center of the 



Ik 

 base of the solid of revolution, and if for convenience — is 



P 

 made equal to c, y must disappear for x=a and therefore 



B=-Ae- 2a/c ; 

 y = A(s- x/c -e- 2a/c s x/c ). 



When a = co, e = and the equation reduces to the sim- 



ple, well-known logarithmic form of uniform resistance. If 

 the origin is taken at the summit of the cone, and both x and y 

 are multiplied by c, the equation may be written 



y=b(e -s). 

 Here b must be determined to correspond to the equation of 

 condition which becomes 



f(y*-y'*) dx=±Vx=min. 



The more b exceeds V therefore, the smaller will be 4& 2 x. Now 



it is well known property of \ that it exceeds its square more 



than any other number, and consequently b=\. Introducing 



this value and restoring c the equation becomes 



— x/c x/c 

 y _ € ' — e ' 



~c~ 2 



