In the mathematical development of the theory it is first necessary to define the coefficient of viscosity. This is done as follows:—If two parallel surfaces AB, CD are separated by a viscous film, and if whilst CD is fixed AB moves in a tangential direction with velocity U, the surface of the film in contact with CD clings to it and remains at rest, whilst the lower surface of the film clings to and moves with the surface AB. At intermediate points in the film the tangential motion of the fluid will vary uniformly from zero to U, and the tangential resistance will be F = µU/h, where µ is the coefficient of viscosity and h is the thickness of the film. With this definition of viscosity and from the general equations representing the stress in a viscous fluid, the following equation is established, giving the relations between p, the pressure at any point in the film, h the thickness of the film at a point x measured round the circumference of the journal in the direction of relative motion, and U the relative tangential velocity of the surfaces,

d (h³ dp) = 6µU dh
dx dx dx

(1)

In this equation all the quantities are independent of the co-ordinate parallel to the axis of the journal, and U is constant. The thickness of the film h is some function of x, and for a journal Professor Reynolds takes the form,

h = a {1 + c sin(θ − φ0)},

in which the various quantities have the significance indicated in fig. 12. Reducing and integrating equation (1) with this value of h it becomes

dp = 6RµUc {sin(θ − φ0) − sin(φ1 − φ0)}
dθ a²{1 + c sin(θ − φ0)}³

(2)

φ1 being the value of θ for which the pressure is a maximum. In order to integrate this the right-hand side is expanded into a trigonometrical series, the values of the coefficients are computed, and the integration is effected term by term. If, as suggested by Professor J. Perry, the value of h is taken to be h = h0 + ax², where h0 is the minimum thickness of the film, the equation reduces to the form

− dp = 6µU + C
dx (h0 + ax²)²(h0 + ax²)³