212 
MR. G. H. DARWIN ON THE STRESSES 
Plate 20, fig. 5, illustrates the results of the representation, the portion of a circle 
marked “mean level of earth” represents a meridional section of the earth; the 
dotted curve marked “ inequality to he represented ” shows the true value of the function 
2exp.[— cos 2 0/ sin 3 a]—/3 0 ; the curve marked “representation” shows the right-hand 
side of (42) stopping with the 12th harmonic; the second curve is the same without 
the 2nd harmonic constituent /3 2 <s 2 , and it is introduced for the reasons explained in the 
discussion and summary at the end. 
The equatorial value of the exponential function is 1‘792, that of the “representa¬ 
tion ” is 1*497, and that of “the representation without the 2nd harmonic” is 1*130. 
The polar value of the exponential is —*3078, that of the “representation” is 
— *0830, and that of “the representation without 2nd harmonic” is +*6516. This 
latter function thus gives us an equatorial continent and a pair of polar continents of 
less height. 
The extreme difference of height in the “ representation ” between the elevation at 
the equator and the depression at the pole is (l*497 + *083)7i or’l *58A I do not exactly 
know the extreme difference in the case where the 2nd harmonic is omitted, because 
I have not traced the inequality throughout, but it is probably about 1*3 or 1 *4A. 
Now let A i be the numerical value, as computed for § 7, of the stress-difference due 
to the harmonic spheroid Then it has been shown above that the stress-difference 
due to the spheroid whose equation is r=a-\-hsi is — 2 (i — l)gwhAi/(2i-{-l). 
Now stress-difference is a non-linear function of the component stresses P, B, T, and 
therefore the stress-difference due to a compound harmonic spheroid is not in general 
the sum of the stress-differences due to the constituent harmonic spheroids. At any 
point, however, where the principal stress-axes are all coincident in direction and where 
all the greater stress-axes coincide, and not a greater with a less, and where T=0, the 
stress-difference is linear and is the sum of the constituent stress-differences. This is 
the case at the equator for the present equatorial continent. 
Hence, if A be the stress-difference at the equator due to the spheroid, 
?-=a+A(/ 3 3 5 2+AS4.+ • • • +A2S12) 
We have 
A=— cjwh [|-/TA.,-f-9A1A4+T'aftAo~b ttAAs+ 10~bAs] * • • ( 43 ) 
In this formula the A/s are the numbers which were computed for drawing Plate 20, 
fig. 4, from the formula (41), namely 
fr'V^Y (rV _3_" 
2(i + 1) 2 + 1 \a) L W " l "(* , -l)(2t+3)_ 
By using these computations I have drawn Plate 20, fig. 6. The vertical ordinates 
are —A - 7 -gwh, and the horizontal are the distances from surface or centre of the 
sphere. 
