OP the earth's axis oe eighke. 39 



a positive value, the pole is shifted in the prolongation of the 

 meridian on which the transfer takes place *. 



11. Prom this formula we can draw several conclusions, of which 

 we may specify the following : — If the latitudes are one north and 

 the other south, the effect is considerable relatively to the amount of 

 matter moved, and is greatest when the transfer is from 45° N . to 

 45° S. (or vice versa) ; and then the right-hand side of equation (4) 

 is 2ma 2 f. 



12. This conclusion is of great importance. If we suppose many 

 masses removed, the numerator of the fraction in equation (3) will 

 merely be the sum of their individual effects ; and as they cannot 

 all be moved from lat. 45° N. to lat. 45° S., their total effect cannot 

 be so great as it would be if they were all concentrated in a point 

 and transferred from lat. 45° IS. to lat. 45° S. ; and consequently, if 

 jj. is the whole mass moved, the effect of the transfer on the position 



* With a view to certain statements in the postscript, it may be well to give, 

 in the form of rules, some of the results that follow from the above formula, in 

 cases where matter is transferred from one point to another along a given 

 meridian : — 



(a) If the matter is transferred from a point short of 45° S. to any point in 

 north latitude, the displacement of the north pole is in the prologation of the 

 meridian ; i. e. it takes place on the meridian whose longitude exceeds that of 

 the given meridian by 180°. 



(b) If matter is transferred from a point in north latitude to a point to the 

 south but still in north latitude, the displacement of the north pole will be as 

 in the last rule (a), provided the sum of the two latitudes exceeds 90°. 



(c) But in the last rule (b), if the sum of the latitudes is less than 90°, the 

 displacement of the north pole will be on the meridian, and not on its pro- 

 longation. 



(d) If matter is transferred from a point in north latitude to a point further 

 to the north, and if the sum of the latitudes exceeds 90°, the displacement of 

 the north pole is on the meridian, as in (c). 



It may be well to add that, if we suppose matter to be transferred along a 

 parallel of latitude, the displacement of the pole is given by the formula 



6 {C-A)=2ma 2 sin 2L sin \l, 



where L denotes the latitude, and I the difference between the longitudes 

 of the points from and to which the transfer takes place. The dis- 

 placement of the pole (north pole if the latitude be north) is on the great 

 circle at right angles to the meridian, midway between the points from and to 

 which the transfer takes place, and in a direction opposite to that of the transfer. 

 Hence we may state the following rule : — 



(e) If the longitude of the middle point of the space over which the transfer 

 takes place is X E., then, if the transfer takes place from west to east in the 

 northern hemisphere, the displacement of the pole is on the meridian of X — 90° 

 E. ; e. g. the tendency of the denudation caused by the Thames, whatever its 

 amount, is to displace the pole on the meridian of —90° E., or 90° W., i. e. 

 towards the north coast of America. 



t It follows from the formula in the last note that, if matter is transferred 

 along a parallel of latitude, the greatest effect is produced when the transfer 

 is in latitude 45° IS. or S., and through a difference of longitude of 180° ; and 

 then the effect is the same as that stated in the text in the case of transfer from 

 45° N. to 45° S. 



