_
/| T23
/
3 /
<----------* 2
T31 1 \
\
_\| T12

The experimenters who have dealt with this question, C.G.M. Marangoni,
van der Mensbrugghe, Quincke, have all arrived at results inconsistent
with the reality of Neumann's triangle. Thus Marangoni says (_Pogg.
Annalen_, cxliii. p. 348, 1871):--"Die gemeinschaftliche Oberflache
zweier Flussigkeiten hat eine geringere Oberflachenspannung als die
Differenz der Oberflachenspannung der Flussigkeiten selbst (mit Ausnahme
des Quecksilbers)." Three pure bodies (of which one may be air) cannot
accordingly remain in contact. If a drop of oil stands in lenticular
form upon a surface of water, it is because the water-surface is already
contaminated with a greasy film.

On the theoretical side the question is open until we introduce some
limitation upon the generality of the functions. By far the simplest
supposition open to us is that the functions are the same in all cases,
the attractions differing merely by coefficients analogous to densities
in the theory of gravitation. This hypothesis was suggested by Laplace,
and may conveniently be named after him. It was also tacitly adopted by
Young, in connexion with the still more special hypothesis which Young
probably had in view, namely that the force in each case was constant
within a limited range, the same in all cases, and vanished outside that
range.

As an immediate consequence of this hypothesis we have from (28)

K = K0[sigma]^2, (49)

T = T0[sigma]^2, (50)

where K0, T0 are the same for all bodies.

But the most interesting results are those which Young (_Works_, vol. i.
p. 463) deduced relative to the interfacial tensions of three bodies. By
(37), (48),

T'12 = [sigma]1[sigma]2T0; (51)

so that by (47), (50),

T12 = ([sigma]1 - [sigma]2)^2 T0 (52)

According to (52), the interfacial tension between any two bodies is
proportional to the square of the difference of their densities. The
densities [sigma]1, [sigma]2, [sigma]3 being in descending order of
magnitude, we may write

T31 = ([sigma]1 - [sigma]2 + [sigma]2 - [sigma]3)^2 T0

= T12 + T23 + 2([sigma]1 - [sigma]2)([sigma]2 - [sigma]3) T0;

so that T31 necessarily exceeds the sum of the other two interfacial
tensions. We are thus led to the important conclusion that according to
this hypothesis Neumann's triangle is ne