/•'. 11 cfa r—JProqf <>f f/<< Earth'* Rigidity. 



389 



which is known as the "equilibrium tide."* In the elastic 



problem, also the globe is supposed subjected to the attraction 



of two half-moons, l>nt the resistance lo deformation is sup- 



d due solely to elastic forces Instead of t<> gravitation. 



tfu str<ss. — Let the earth's center of inertia be 

 taken as the center of coordinates; let r be the distance of anj 



point in Bpace from this origin and let 3 be the angle which r 



makes with the X axis. Let also fi and x be the distances of 

 the point from the two portions of the moon, and let M be 

 the mass of the moon and I) its mean distance. Then, by the 

 law of gravitation, the forces acting on the point are 



and here 



-— 2 and 

 2x 



* f =D»(l +~-% cos 3) ; yu'=D' (1 + £ + ^ c 



D 



D 3 ' D 



These forces can of course be resolved in any direction by 

 simple projections. The resulting formulas can also be sim- 

 plified without substantial loss of accuracy by neglecting powers 

 of r/D higher than the first. The distance of the moon is 

 60*3 times the radius of the earth, and the highest value of 

 r*/ D 2 which can occur in this discussion is therefore only 

 1/3,636. 



If the forces are resolved into F n the component coinciding 

 in direction with the radius vector, and F t the component at 

 right angles to the radius it will be found that 



"Mr "\[r 



F r = -=p (3 cos a 3-1) and F t = — 3 sin 3 cos 3. 



Or if the forces are resolved into components X. V, Z in the 

 directions of the axes (reckoned as usual per unit volume) 



* This system of '• moon and anti-moon'' is that employed by Laplace, Thom- 

 son and others. Prof. 6. H. Darwin has given a far more elegant method of 

 dealing with the equilibrium tide (Encycl. Brit., Article Tides. 1888); but though 

 his treatment is very simple, it is less adapted to a paper like this, from which it 

 seems best to exclude the potential. 



