DUE TO THE WEIGHT OF CONTINENTS. 
211 
and are those exhibited graphically in fig. 4; they are here given as a means of 
comparison with the numbers corresponding to latitudes 6° and 12°. 
The result to be deduced from this table is that the lines of equal stress-difference 
are very nearly parallel with the surface, and that it is for all practical purposes 
sufficient to know the stress-difference immediately under the centre of the continents. 
We have already seen in § 6 that for harmonics of infinitely high orders the line’s of 
equal stress-difference are rigorously parallel with the mean surface. 
§ 8. On the stresses due to the weight of an equatorial continent. 
The actual continents and seas on the earth's surface have not got quite the regular 
wavy character of the elevations and depressions which have been treated hitherto. 
The subject of the present section possesses therefore a peculiar interest for the 
purpose of application to the earth. Had I however foreseen, at the beginning, the 
direction which the results of this whole investigation would take, it is probable that 
I might not have carried out the long computations which were required for discussing 
the case of an isolated continent. But now that that end has been reached, it seems 
worth while to place the results on record. 
The function exp.[— cos ' l 6f sin 3 a] (where-d is colatitude) obviously represents an equa¬ 
torial belt of elevation, and I therefore chose it as the form of the required equatorial 
continent. This function has to be expanded in a series of zonal harmonics in order 
to apply the analytical solutions for the stresses produced by the weight of the 
continent. 
It is obvious by inspection that the odd zonal harmonics can take no part in the 
representation of the function, and it was on this account that I have above only 
worked out the cases of the even zonal harmonics. 
The multiplication of this function by the successive Legendre’s functions, and 
integration over the surface of the sphere, are operations algebraically tedious, and 
wholly uninteresting, and I therefore simply give the results. 
I find then that if a =10°, and 
>*= sin* 0—~ sin* 3 0 cos 3 0 + ^ 
21 
4! 
sin* 4 0 cos 4 0—kc. 
Then 
where 
2e cos-*/sm*«_^ 0 ==^ 2 s 2 +^ 4 5 4 +^ 6 5 a +^ 8 5 8 +^i oSlo 4.^i 3 ^. , . . 
f3 0 =-3Q78, &=-3673, &=*3339, 0 6 =‘2829, 0 8 =*2252, 0 1O =-1688, > ’ 
&*=-1193 
(42) 
A, is P l| t 011 the left-hand side in order that the mean value of the function may be 
zero. The 0 s obviously decrease very slowly, and as I stop with the 12th harmonic, 
the representation of the function is very imperfect. 
2 E % 
