relation to the Constitution of the Earth? 8 Crust. 17 



x f 77 r~ ~> — 7i 1 ^9 u(h + k + t\ 



= {a—/j,)a\u-\-h + k + t- \fu 2 +(h + k + t) 2 + -^ — z L 



_(h ± k ± ty_( u +h+k _ ^ Hh+kf ^JJ^^M^\\ 



c \ 2c c J) 



= (o— a*)« t- \fu 2 + (h + k + t)*+ ^u 2 + (h + k) 2 +^ 



2(A + fe> + £ \ 



So that the whole vertical attraction of the cap-sector at the 

 extremity of its axis is 



f , /-o — tt u h h 2 > 

 pa < u + h— Vw -H h + o f 



}• 



2c 

 2(A-r&> + * 2 

 c 



But the condition of hydrostatic equilibrium gives 



ph = (cr-fi)(l — r)t; 

 and it will appear that fractional quantities in - may be 

 neglected. ,: P h={<r-f*)L 



Hence, considering an elementary sector of axial angle Sot, 

 its attraction will be 



p 8u(u- Vu 2 + h 2 ) + (<T-jj,)$u(</u 2 + (h + k + t) 2 



- Su* + (h + ky), 

 the terms in h, t, uh 9 and ut cancelling, by the condition of 

 equilibrium. Thus the first part of Young's correction dis- 

 appears. 



The above expression affords a simple criterion of the 

 amount of the vertical attractive force at the extremity of the 

 axis, arising from the disturbance of gravity by the matter 

 constituting a cap-sector, on the supposition of hydrostatic 

 equilibrium. It is proportional to the difference between the 

 extreme distances of the upper and lower edges of the root, 

 multiplied by the relative density thereof, diminished by the 

 difference of the distances of the upper and lower extreme 

 edges of the visible mass, multiplied by its density. But 

 when the distances are great, the thickness of the visible 

 mass being small compared with that of the root, the 

 attraction may be regarded as practically proportional to the 

 difference of the distances of the upper and lower edges of 

 the root. It follows that the attraction is less the longer the 

 sector, and becomes insensible when that is very long. 



Phil. Mag. S. 5. Vol. 22. No. 134. July 1886. C 



