170
VI
ON HARDNESS
For the purpose of what follows it is convenient to
introduce an auxiliary angle by the equation
and then
COS T =
A-B
A+B'
2A = (P11+ P12 +P22 + P22)sin²
2
2B = (P11 + P12 + P21 + P22) Cos³·
2
We shall introduce these values into the equations for
a and b, and at the same time transform the integrals
occurring there by putting in the first u=b222, in the second
u = a²². Denoting the ratio b/a by k we get
1
2)
dz
=
4π P11+ P12 + P21+ P22 sin??
D1 + D2
√(1+k²x²)³(1 + z²) — 3p
dz
4π P11+ P12+ P21+ P22 cos2
1
√(1+1)*(1
3p
(1+z2)
D1 + D2
2
>
2
Dividing the one equation by the other we get a new one,
involving only k and T, so that k is a function of alone;
and the same is true of the integrals occurring in the equa-
tions. If we solve them by writing
μ
8
3p(D1 + D2)
8(P11+P12+ P21+ P22) '
3
3p(D1 + D2)
= v
8(P11+P12+ P21+ P22)'
=
then μ and depend only on T, that is on the ratio of the
axes of the ellipse e constant. The integrals in question
may all be reduced to complete elliptic integrals of the first
species and their differential coefficients with respect to the