1876.] On determining the Depth of the Sea. 317 



The following Papers were read : — 



I. " On determining the Depth of the Sea without the use of the 

 Sounding -line, and on an Attraction-meter." By C. William 

 Siemens, D.C.L., F.R.S., M.I.C.E. Received January 20, 

 1876, 



(Abstract.) 



The author starts with the proposition that the total gravitation of the 

 earth, as measured on its normal surface, is composed of the separate 

 attractions of all its parts, and that the attractive influence of each equal 

 volume varies directly as its density and inversely as the square of its dis- 

 tance from the point of measurement. 



The density of sea-water being about 1-026, and that of the solid con- 

 stituents composing the crust of the earth about 2*763 (this being the 

 mean density of mountain-limestone, granite, basalt, slate, and sandstone), 

 it follows that an intervening depth of sea-water must exercise a sensible 

 influence upon total gravitation if measured on the surface of the sea. 



The amount of this influence is proved mathematically in considering, 

 in the first place, the attractive value of any thin slice of substance in a 

 plane perpendicular to the earth's radius, supposing the earth to be a 

 perfect sphere, unaffected by centrifugal force, and of uniform density. 



The attraction of each such slice is shown to be represented by the 

 expression 



ddA l = 2-ndh sin a da , 



in which h represents the vertical distance to the slice from the point of 

 attraction, and A x the total attraction of the slice. 



In integrating this expression between the limits h and and a and 0, 

 this gives 



in which, for small values of h, the factor \ /——- may be neglected, when 



V 2E/ 



the formula assumes the form 



A 1 = 2 7 r^ 



in which A x represents the total attraction down to the depth h. 



If A represents the total attraction of the earth, the following propor- 

 tion obtains, 



A x __ ZttTi 



A |Ett ' 

 or 



ft-fR- 



It follows that if sea-water was without weight, the total attraction 

 of the earth, as measured upon the sea-surface, would diminish in the pro- 

 portion of the depth to §R; but taking the weight of sea- water into 



