324 



Prof. G. H. Darwin. 



[Apr. 30, 



plane, and the third is perpendicular thereto. Moreover the greatest 

 and least stress-axes are those which lie in that plane." 

 And in a foot-note on p. 200 — 



" It is easy to see that if a viscous sphere be deformed into the 

 shape of a zonal harmonic, the flow of the fluid must be meridional r 

 and from this we may conclude that in the elastic sphere the plane of 

 greatest and least principal stresses must be also meridional. This 

 has already been assumed to be the case in the present paper." 



As one of the examiners for the Smith's Prizes at Cambridge, I 

 have had placed before me an essay by Mr. Charles Chree, of King's 

 College, in which he considers, amongst other points, the difference 

 of principal stresses in an elastic sphere strained under the influence 

 of the forces due to a potential expressed by the second zonal har- 

 monic. In this essay Mr. Chree has pointed out that the conclusion 

 thus arrived at by general reasoning is erroneous. His analytical 

 treatment of the problem is entirely different from mine, and I cannot, 

 therefore, avail myself of his actual work in amending the error 

 which he has pointed out and corrected. 



It is clear that, in the limiting case of the zonal harmonic where 

 the surface becomes a series of parallel mountains and valleys on a 

 flat surface, the principal stress parallel to the mountains must be 

 zero, and the above reasoning has led to a correct conclusion. 



But in the case of the second zonal harmonic, with either excess 

 or deficiency of matter at the pole, there is a tendency for the equa- 

 torial regions to be either squeezed out or crushed in. Now an out- 

 ward squeeze necessitates that the greatest pressure shall be perpen- 

 dicular to the meridian, and this is contrary to the general conclusion 

 quoted above. My error lay in overlooking this outward or inward 

 tendency in the equatorial matter. 



The conclusion is therefore wholly right in the case of the moun- 

 tains and valleys, and at least partially wrong in the case of spheroidal 

 deformation of the globe. 



The data for examining into this question rigorously are given in 

 my paper, and the best way of treating the matter is to rewrite § 5 

 on — 



The State of Stress due to Fllipticity of Figure or to Tide- generating 



Forces. 



When the effective disturbing potential Wi is a solid harmonic of 

 the second degree, the solution found will give the stresses caused by 

 oblateness or prolateness of the spheroid. It will also serve for the 

 case of a rotating spheroid with more or less oblateness than is 

 appropriate for the equilibrium figure. When an elastic sphere is 

 under the action of tide-generating forces, the disturbing potential 



