Littlehales — Form of Isolated Submarine Peaks. 15 



Art. III. — The Form of Isolated Submarine Peaks; by 

 G-. W. Littlehales, U. S. Hydrographic Office. 



Theoretically the form of an isolated submarine peak 

 would be that of a solid of revolution in which the crushing 

 strength of any section is equal to the combined weight of the 

 portion of the formation above that section and of the superin- 

 cumbent body of water. Let y denote the radius of any sec- 

 tion, and x its distance from the top of the formation. Let K 

 denote the coefficient of crushing strength of the material 

 composing the formation ; 3, the weight of a unit of its vol- 

 ume ; and 3', the weight of a unit of volume of sea- water. 



Assuming that the top of the formation just reaches to the 

 surface of the ocean, 



7tdl y*dx = the weight of the formation above any section 

 " whose distance from the top is sc, 



27td' /y.x. dy = the pressure of the water upon the formation 

 J above any section whose distance from the top is x, 



7rKy 2 = the strength of any section to resist crushing. 



Then 



7rdfy' i dx + 27rd , /y. x> dy = 7tKy* + C (1) 



in which C is a constant representing the excess of crushing 

 strength in any section above what is necessary to withstand 

 the pressure caused by the weight of the formation and the 

 weight of the superincumbent body of water. 

 By differentiation, equation (1) becomes 



7tSy 2 . dx + 27td'y. x. dy = 27rK . dy 



d dx dy 



or 



2 (K-d'x) ~~ y 



6 dx dy 



By integration, equation (2) becomes 



^log(x-j) = Iogy 



or x = j-, + £ 



k t lo s y ( 3 ) 



in which e is the base of the Naperian system of logarithms. 



