Rigidity of the Earth. 505 



of the tides were neglected, a force which has a potential 

 <j a) 2 sin #cos Ol —jk ) 



would raise a tide whose height (from low to high water) 



would be -x — sin#cos0. Now since g = 290aco 2 , we find that 



the height of this tide is ^~ — -=0*32 inch if we take 



6 = 0" -15. This would he the actual difference of level of 

 the ocean in latitude 45° due to the positive tide which 

 results from the shifting of the earth's axis of rotation. For 

 the circle of low water touches the parallels of latitude 45° 

 north and south, and the crests are in the same latitude. 

 Thus on each of these parallels there is high and low water 

 separated by 180° of longitude. Consequently the change of 

 level at any place in latitude 45° would be, on the preceding 

 assumptions, the whole height of the tide, 0*32 inch. 



The negative tide lags behind the positive tide by half a 

 period, and therefore its depression coincides with a point of 

 low water of the positive tide. Thus the whole change of 

 level due to the two 45°-tides would be 064 inch. 



But 64 inch will not be the actual change of level if the 

 earth yields to tidal forces. According to W. Schwcydar, 

 whose work has been referred to under ; Tidal Observations ' 

 in this paper, the actual height of a long period tide is 0"02 

 of its theoretical height. This makes the change of level due 

 to the 45°-tides become nearly 0*4 inch. This is a measur- 

 able quantity, and if a tide of this height were found with a 

 period of 430 days it would be strong confirmation of the 

 result deduced from observations of latitudes. 



It is useful to compare this tide with one of the long- 

 period tides, for example, the monthly tide. It is shown in 

 Art. 848 of Thomson and TahVs ' Natural Philosophy ' that 

 the mean change of level at the equator due to the monthly 

 tide would be 87 inch on a rigid earth. Thus the change 

 of level due to the shifting of the earth's axis of rotation, 

 assuming that this axis describes a cone in the earth whose 

 semi-vertical angle is //# 15, is nearly J of the change of level 

 at the equator due to the monthly tide. 



These 45° tides themselves would, in the course of time, 

 bring the axes of rotation and of figure into coincidence, even 

 without the assistance of the lunar and solar tides. But the 

 rate of decrease of the angle between these axes differs so 

 greatly for different assumptions concerning the effect of tidal 

 friction that it does not seem very profitable to try to draw 

 any conclusions from the results. 



