VISCOSITY AND FLUIDITY 89

whether <p must be invariably greater than, equal to, or less
then V'- Multiplying Eq. (25) by unity, we obtain

Since 6 = 1 — a,

2a(a - 1W2 + a(l - a)

Discarding the known roots, a = 0 and a = 1, we get

<£>12 — 2<pi<p2 + <P22 ^ 0,

which is a perfect square and therefore must be positive. Hence,
when (pi is equal to <p% (p is equal to <p', but for all other values
<p must be greater than <?'. Our conclusions may be stated as
follows:

1. The viscosity of a thorough mixture of chemically indifferent
fluids must always be less than would be expected on the assump-
tion that viscosities are additive, but this inequality will approach
zero as the difference between the viscosities of the components
approaches zero.

2. So, on the other hand, the viscosity of an emulsion must be
greater than that of a perfect mixture of the same composition,
because in emulsions the viscosities tend to become additive.

Equation (25) may be expressed in the form

<f> = 91 +0?2 — <pi)b, (29)

where <pi and <pz are constant and <p and b are variable. The'
corresponding viscosity equation is

(30)

n = , - TT

<Pl + (<f>2 — <Pl)0

It is important to note that Eq. (29) is the equation of a
straight line, but that Eq. (30) is the equation of a hyperbola.
If we replace <pi +(<p2 — ^1)62 by (&i — <PI)W, where

we get

which is the equation of an equilateral hyperbola, whose X-axis is