DUE TO THE WEIGHT OF CONTINENTS. 191. 
/p =1 
cP Wi 
clx 3 
-3 iWi 
IT= ■ 
ffv-a+zy) 
cP Wi 
dxdz 
8 
CO 
1 
dW,\ j 
^ dx ) J 
where /= 2 (i-fi 1 ) 3 +1. 
The expressions for Q, It, S, TJ may be written down from these by means of cyclic 
changes of the symbols. 
These are the required expressions for the stresses at any point in the interior of 
the sphere. 
In order to find the magnitude and direction of the principal stress-axes at any 
point it would be necessary to solve a cubic equation. The solution of this equation 
appears to be difficult, but the special case in which it reduces to a quadratic equation 
will fortunately give adequate results. It may be seen from considerations of symmetry 
that if Wi be a zonal harmonic, two of the principal stress-axes lie in a meridional 
plane and the third is perpendicular thereto. Moreover the greatest and least stress- 
axes are those which lie in that plane, and the mean stress-axis is that which is 
perpendicular thereto. If this is not obvious to the reader at present, it will become 
so later. 
I shall therefore take W) to be a zonal harmonic, and as the future developments 
wifi be by means of series (which though finite will be long for the higher orders of 
harmonics) I shall attend more especially to the equatorial regions of the sphere. 
2. The determination of the stresses when the disturbing potential is an 
even zonal harmonic. 
If 0 be colatitude the 
function of order i is 
expression for a zonal surface harmonic or Legendre’s 
cos* 6 - 
i(i 
t) s\ • o n i 'it% 1)(^ 2)G o) ; & n • An 
4.(1 ly cos " 6 sm " eH - 48(21)2 — cos 0 sm e ~ 
or if we begin by the other end of the series, and take i as an even number, the 
expression is 
(-)U 
2 W !}1 
sm 
1 2 \ S4n * 3 ^ cos3 sin*“ 4 0 cos 4, 0- 
This latter is the appropriate form when we wish to consider especially the equatorial 
regions, because cos 0 is small for that part of the sphere, 
There is of course a similar formula when i is odd, but of this I shall make no use, 
Now let p 3 =?/ 3 +z 3 , so that sin 0=p/r i cos 0=z/r> 
