ME. G. H. DAE WIN ON THE STEESSES 
§ 7. On the stresses due to the even zonal harmonic inequalities. 
Having considered the two extreme cases where i is 2, and infinity, I pass now to 
the intermediate ones. As the odd zonal harmonics were not required for the investi¬ 
gation in the following section I have only worked out in detail the even ones. 
The surface of the sphere is now corrugated by a series of undulations parallel to 
the equator, and the altitude of the corrugations increases towards the poles. The 
form of the undulation in the neighbourhood of the equator is exhibited in Plate 19, 
fig. 3. 
The stress-difference is as before given by 
To form this expression the series in (22) for It must be subtracted from the series 
for P. Since the C’s and Z)’s of Table II. have always the opposite signs from the A ’s 
and B s of Table I., this algebraic subtraction becomes a numerical addition of the 
numbers in these two tables. 
The results are given in the following table. 
Table IV.—The coefficients for expressing P—R. 
i 
O 
1 
A 2 -0 2 
A.-C, A g -C 6 
i? 0 i) 0 
-®2 ^2 
^6-^6 
2 
-2-2105 
-2-8421 
+ 2-5263 
4 
-4-3137 
+ 2-7451 
+ 7-8431 
+ 4-3922 
-7-5294 
6 
-63636 
+ 42 
+ 35-6364 
-14-2545 
+ 6-4 
-48-8727 
+ 13-9636 
8 
-8-3926 
+ 139-1411 
-38-8712 
-168-2061 
+ 8-4137 
-148-0806 
+ 188-4663 
-21-5390 
10 
-10-4115 
+ 318-2304 
-633-7449 
-478-0247 
+ 10-4252 
-329-2182 
+ 965-7051 
-491-6324 
12 
-12-425 
+ 603-292 
-2623-009 
+ 552-212 
+ 12-434 
-616-315 
+ 3243-765 
-3806-018 
Then we have 
P—R=7*'cosrf[(rf 0 —C 0 ) + (rf 3 —C 3 ) tan 3 Z+ . • •] 
-f-aV -3 cos* -3 l[_(B 0 —D Q )-\-(B 2 —D 2 ) tan 2 £-J- . . .] 
The materials for computing T have been already given in Table III. 
The series for P —R and for 2T should now be squared and added together, but the 
result would be so complex that it is not worth while to proceed algebraically. 
At the equator T=0, and A=P —R, and P—R reduces to only two terms, 
whatever be the order of harmonic. 
By reference to (23) and (24) we see that at the equator 
