338 G. F. Seeker— Proof of the Earth's Rigidity. 



logical circles to admit the necessity of a result the demonstra- 

 tion of which scarcely anyone — perhaps no one — generally 

 classed as a geologist can follow critically. 



Possibility of a simplification. — It occurred to me that, 

 since ever the result of the Lame-Thomson investigation is 

 only an approximation for real (or finite) strain there might 

 be some simpler way of reaching another approximation which, 

 though less accurate, would be sufficiently close for a demon- 

 stration of the rigidity of the earth. Thomson's result is that 

 the ellipticity of the elastic sphere, under the limitations men- 

 tioned above, is T 5 g of a certain constant. If one could show 

 that this ellipticity were, at all events, greater than J of this 

 same constant, it will appear by an examination of Thomson's 

 argument that substantially the same results would follow. 



I have succeeded in proving this inequality by a method so 

 simple that no mathematics beyond plane trigonometry is need- 

 ful to follow it. Besides presenting this proof in the most ele- 

 mentary possible manner to my colleagues, I shall take the oc- 

 casion to follow out Thomson's argument in the way which 

 seems best adapted to geological readers, for it appears to me 

 certain that on some points he has been very generally misun- 

 derstood. 



General character of the earth's elastic strain. — The prob- 

 lem ©f the deformation of an elastic sphere by the attraction of 

 an external body like the moon is closely analogous to the 

 simplest tidal problem. Suppose that the earth were always 

 to present the same face to the moon, just as the moon presents 

 (nearly) the same face to the earth. Then the waters would 

 be drawn upward toward the moon to a definite ellipsoidal 

 surface capable of accurate computation. This ellipsoid would 

 have three unequal axes, of which the longest would be di- 

 rected toward the moon, while the polar axis would be the 

 shortest. The difference between the polar axis and the shorter 

 of the equatorial axes would be due to the rotation of the 

 earth-moon system, and but for this rotation the two bodies 

 would fall together. This rotation, however, produces an ellip- 

 tical flattening of the globe, while the attraction of the moon 

 produces a distinct but superimposed elongation of the globe. 

 It is desirable to separate these two effects and this can be 

 done by the following device. The earth may be supposed at 

 rest in space and subjected to the action of two bodies fixed at 

 equal distances from it in opposite directions. If each of these 

 bodies has half the mass of the moon, and if the distance of 

 each of them from the earth's center is equal to the moon's 

 real mean distance, then the deformation will be sensibly the 

 same as that which the real moon would produce irrespective of 

 the effects of rotation. It is the tide found on this hypothesis 



