282 Archdeacon Pratt on the Thickness of 



two unsymmetrical hemispheroids into which the plane yz sepa- 

 rates it. The disturbing forces parallel to y and z will, on the 

 other hand, produce compression in directions parallel to those 

 lines. These also will produce two couples, of small moment, 

 tending to make the earth rotate round diameters perpendicular 

 to their planes and lying in the planes xz and xy respectively. 



4. It is not difficult to calculate these forces and their mo- 

 ments about the axes. I will do this, first, supposing the earth's 

 mass to be homogeneous. 



Let / be the longitude of the sun, o> the obliquity of the 

 ecliptic, a the mean radius of the earth, e the ellipticity. Then 

 the equation to the surface is 



# 2 + ?/ 2 + 2 2 = # 2 (1 + e) — 2e (x sin I sin to + y cos I sin to + z cos to) 9 



= a 2 (l + e) -2e(J)x + 'Ey + Yz) 2 i 

 where 



D= sin /sin a), E== cos /sin ft), F=cosft); 



the square of the ellipticity I shall neglect ; 



.-. #=-2D(Ey + F < ?)e + Vd?{l + e) -y*-s*-2e{fy + Yz)*. 

 It will be convenient to put for a time 



y + 2'E¥z6=Y ) and a(l+e(i-F 2 )) = «. 

 Then the values of x will be 



a?=-2D(Er + F,?)e±(l+E 2 e) Vl + 2e(F 2 -E 2 )(a 2 -* 2 )^-Y 2 , 



and the limiting values of Y in the plane yz will be 



Y=±(H-e(F 2 -E 2 )) VV^*, 



and the limiting values of z will be +a. 



As it will be useful, I will write down the value of x* : 



a? 2 =(l + 2eF 2 )(a 2 -^ 2 )-(l + 2eE 2 )Y 2 + 4D(EY + F^)A/a 2 -^ 2 -Y 2 . 



The whole force parallel to x on the hemispheroid nearest the 

 sun 



= 2fz f f J pdz dydx.x=fjip f J dz dY x 2 ~ | fipircft, 



neglecting the small term in e compared with the first. This is 

 the force of tension sustained at every instant by the section of 

 the earth by the plane yz } there being an equal force drawing 

 the opposite half in the opposite direction. The force of com- 

 pression perpendicular to the planes xz, xy will be half this. 



5. I must now find the moment of the forces parallel to x 

 about the axes of y and z. I will here once for all say that 

 those moments will be reckoned positive which tend to turn the 

 earth about the axis of x from the axis of y to the axis of z, about 

 y from z to x, about z from x to y. 



