1877.] 



The Rev. S. Haughton on Physical Geology. 



51 



IV. " Notes on Physical Geology/ 5 By the Rev. Samuel 

 Haughton, M.D. Dublin, D.C.L. Oxon., F.R.S., Professor of 

 Geology in the University of Dublin. Received February 15, 

 1877. 



No. I. Preliminary Formulas relating to the internal change of position 

 of the Earth's Aocis, arising from Elevations and Depressions caused 

 by Geological Changes. 



1. If the earth's surface be an ellipsoid of revolution, whose moments 

 of inertia round the polar and equatorial axes are C and A, and if /j. be 

 the mass of a mountain placed on any meridian, with coordinates z, x, 

 it is required to find the change of position in the earth's axis caused 

 by the addition of the mass \x (supposed in the first instance to be placed 

 upon the earth ah extra). If X be the latitude on which fi is placed, we 



have tan \=- ; and if 6 be the angle made with the earth's axis by any axis 

 x 



in the meridian of fi, if I be the total moment of inertia round this 

 axis, we have 



I = Asin 2 0+Ccos 2 



+ju(Vcos 2 6-\-z 2 sin 2 6— 2xz sin cos 6) . . . . (1) 



The new axis of rotation is that which makes 



1= maximum, or dl=0, 

 from which we find, after some reduction, 



.tan 20=-^ — 



(C-ahk* 2 -* 2 ) 



(2) 



If we make = maximum, or dd=0, we ascertain the position in 

 which the mass fx must be placed so as to produce the maximum shift in 

 the position of the earth's axis. 



Differentiating (2), we find 



x{(C-A) + fi(x 2 + z 2 )}dz+z{(C~A)-fjL(x 2 + z 2 )}dx=0; 

 and from the equation of the ellipse 



c l a 2 



we have 



zdz xdx 



c-' + a 2 



e2 



