VI
169
ON HARDNESS
Finally, it is easily shown that the very essential in-
equality, which must be fulfilled outside the surface of
pressure, is actually satisfied; but I omit the proof, since it
requires the repetition of complicated integrals.
Thus our formule express the correct solution of the
proposed problem, and we may use them to answer the chief
questions which may be asked concerning the subject. It is
necessary to carry the evaluation of the quantities a and b a
step further; for the equations hitherto found for them cannot
straightway be solved, and in general not even the quantities
A and B are explicitly known. I assume that we are given the
four principal curvatures (reciprocals of the principal radii of
curvature) of the two surfaces, as well as the relative position
of their planes; let the former be P1 and P12 for the one
body, P21 and P22 for the other, and let w be the angle
between the planes of p₁ and of Pa
P11
Let the p's be reckoned
positive when the corresponding centres of curvature lie
inside the body considered. Let our axes of xy be placed so
that the az-plane makes with the plane of p₁ the angle w', so
far unknown. Then the equations of the surfaces are
2z₂ =
22₁ = P₁1(x cos + y sin w')² + P12(y cos w' - x sin w')²,
= -
-
P21 {x cos (w' — w) + y sin (w' — w)}²
1
-
— P22 {y cos (w' — w) — x sin (w' — w)}².
The difference z₁-z₂ gives the distance between the surfaces.
Putting it = Ax² + By², and equating coefficients of x², xy, y²
on both sides, we obtain three equations for w', A and B;
their solution gives for the angle w', which evidently de-
termines the position of the axes of the ellipse of pressure
relatively to the surfaces, the equation
for A and B
2(AB) =
-
tan 2':
=
(P21-P2)sin 2w
P11-P12+(P21-P22) cos 2w'
2(A+B) = P11+ P12 + P21 + P22,
- √(P11-P12)²+2(P11-P12)(P21 - P22)Cos 2w+(P21 − P22)²*