330 Mr. J. J. Waterston on the Integral of Gravitation, 



§1- 

 The Integral of Gravitation is a function of Space. 



Suppose a central homogeneous globe to augment in bulk by 

 the descent of similar matter from an infinite distance in radial 

 directions all round ; each descending element, on arriving at the 

 sui'face of the globe, presents itself chai'ged with a certain amount 

 of mechanical force equivalent to the square velocity with which 

 it impinges. If we confine our attention to the centripetal influ- 

 ence of the' original central globe only, the square velocity of the 

 descending element diminishes in the inverse ratio of the radius 

 of the augmenting globular mass ; for it is upon the surface of 

 this that impact takes place, and the matter that has been added 

 to the original globe is assumed not to augment the centripetal 

 force acting upon the descending matter. 



Taking for standard unit the square velocity generated by 

 falling through the radius of the globe with the force of gravity 

 uniform as at surface, and computing the integral mechanical 

 efi"ect between the original surface of the globe and any other 

 spherical surface external to and concentric with that surface, it 

 is found to be equivalent to the product of the standard square 

 velocity by a mass of matter that would cover a surface equal to 

 a great circle of the outer sphere, minus a great circle of the 

 central globe, to a depth equal to the radius of the globe. This 

 ratio is equal to the ratio of the square of the cube root of the 

 space between the concentric spherical surfaces, and continually 

 approaches the ratio of the surface of the outer sphere. 



If the matter composing a planetary globe is assumed to have 

 originally descended from space, and to have become centrally 

 collocated in successive layers, it may be viewed with refer- 

 ence to its gravitation integral (1) as having accomplished work, 

 (2) as having the faculty of accomplishing work, of generating 

 force. The following are a few theorems developing the quan- 

 titative relations. 



1. The work accomplished, or the mean square velocity of the 

 molecules of a planetary globe acquired by the centripetal precipi- 

 tation of the matter of that globe from an infinite distance, is equal 

 to f the square velocity acquired by a body falling through radius 

 of the globe with the uniform force of gravity at its surface. 



The following may serve to illustrate this. 



If as much matter (iron) descended to the earth from plane- 

 tary space as would cover the whole surface to a uniform depth 

 of 2^0*^ ^^ ^^^ inch, it would, on entering the atmosphere with 

 the mean velocity of 20 miles per second, generate as much heat 

 as the whole atmosphere contains. The steps of the computa- 



