38 J. P.TWISDEN ON POSSIBLE DISPLACEMENTS 



6. In applying this formula to the question proposed in art. 1 we 

 observe that d will have nearly its largest value when a is 45°; we 

 will therefore take this value instead of 20° for the obliquity of the 

 belt, and we will take the height of the elevation to be h ; we then 

 plainly get 



tan 2d = 2 ™; (2) 



a 



or, if we take h to be |- of a mile, 6 is an angle of about 9' 17". The 

 answer to the question is, therefore, that the axis of figure would be 

 displaced through an angle of less than one sixth of a degree, and 

 the displacement would take place, not on the meridian of Greenwich, 

 but on its prolongation, i. e. on the meridian of a station in long. 

 180°. It is plain from equation (2) that, to produce a displacement 

 of as much as 1°, h (the height of the elevation) would have to be 

 more than 5 miles. 



7. These results seem conclusive in regard to the particular method 

 of elevation suggested in the extract. But the question may be 

 fairly asked whether elevations and depressions of some other kind 

 might not produce the desired displacement? If any particular 

 kind of elevation or depression is suggested, it can be treated by a 

 method resembling that already employed ; but when the question 

 is asked in a general form, of course a specific answer is not possi- 

 ble ; still some general results may be arrived at which ought at 

 least to be borne in mind by those who speculate on this subject. 



8. Let us suppose the axes of z and x to be those with reference 



to which C and A are taken ; then, if we assume that a mass of matter 



(m) is transferred in the plane of zx from a point whose coordinates 



are (z, x) to a point whose coordinates are (z', x), it is easily shown 



that the displacement of the axis of figure (6) is given approximately 



by the formula 



2m(z'x' — zx) 

 tan 20= K C _ A ~ J ; (3) 



so long as the matter moved is but a fractional part of the whole 

 equatorial bulge, we may safely reason on this formula. 



9. Now, if the matter is transferred from one point to a neigh- 

 bouring point, z'x' will not differ much from zx, and consequently 

 tan 20, and therefore 6, will be but small. This result is important 

 and is general : e. g. matter transferred from a mountain-range into 

 a neighbouring sea, molten matter transferred from an underground 

 region to a neighbouring region above or below ground would pro- 

 duce but little effect on the earth's axis of figure, even if the 

 whole amount of matter transferred were large. 



10. Again if we suppose the matter to be transferred from latitude 

 <p to latitude (j> r , we may write the equation (3) as follows, 



(C - A) tan 2Q~2md i sin (^' — 0) cos (<p' + <f>), . . . . (4) 



with sufficient accuracy for present purposes. If this equation gives 





