202 
MR, G. H. DARWIN ON THE STRESSES 
Then substituting for A, B, C, D, E their values from Tables I., 11, ffl. } and 
making some simple reductions, we find 
P—R=tV[ 8 (a 3 —? * 3 ) — r 2 cos 2 ff] 1 
2T=^ S )" sin 20 J.( 34 > 
Therefore by (29) in the present case, the stress-difference 
A =■i^\/64(a 3 —I6r 2 (<x 3 — r 3 ) cos 20 .(35) 
In order to find the actual value of A in any special case we shall have to multiply 
(3o) by appropriate factors \ the factors will be determined below. For the present it 
w iH be more convenient to omit the factor -pj, and to reintroduce it along with these 
other factors. 
The formula (35) enables us to determine the distribution of stress-difference 
throughout the sphere in the cases to which this section applies. 
The curves of equal stress-difference are given by the equation 
64(a 3 —r 3 )+r 4 — 16r 3 (a 3 — r 2 ) cos 20=ix constant 
The stress-difference at any point on any equatorial radius (for which 6=\tt) is 
clearly given by 8 a 2 —7 r 3 , and along the polar radius (for which 0=0) by 8a 2 —9r 3 . 
From this result it is clear that the stress-difference vanishes at that point on the 
polar radius for which r==fcH/2=‘9425a; this is the only point within the whole 
spheroid for which it vanishes. 
When r=a the stress-difference is equal to a 3 , from which we obtain the remark¬ 
able result that the stress-difference is constant all over the surface. When r= 0, 
it is equal to 8a 3 , which is eight times the surface value. 
By means of arithmetical calculation and graphical interpolation I have drawn 
fig. 1 ; it shows the curves of equal stress-difference throughout a meridional section 
of the spheroid. The numbers written on the curves represent the values of stress- 
difference when the radius of the sphere is unity and when the factors above referred 
to are omitted (see Plate 19, fig. 1). 
The point marked 0 is that in which the stress-difference vanishes. Round this 
point are drawn two dotted curves along which the stress-differences are \ and f 
respectively. The remainder of the curves are drawn for equidistant values of stress- 
difference, and are marked 1, 2, 3 ... 8. The curve marked 1 is singular, for the 
whole of the surface forms one branch of it, whilst there is another branch which runs 
below the surface from the polar axis and then rises to the surface at the point where 
cos 20= — that is to say, in lat. 41° 25'. Near the centre the curves are approxi¬ 
mately circular, and they become somewhat like ellipses as we recede from the 
centre. 
