which is necessary for equilibrium, must therefore be provided by the tensions existing in the _other two_ surfaces of contact. In short, if we could imagine a single particle placed at the very point of contact, it would be drawn upon by three different forces, whose directions would lie in the three surface planes, and whose magnitude would be proportional to the specific tensions characteristic of the two bodies which in each case combine to form the “interfacial” surface. Now for three forces acting at a point to be in equilibrium, they must be capable of representation, in magnitude and direction, by the three sides of a triangle, taken in order, in accordance with the elementary theorem of the Triangle of Forces. So, if we know the form of our floating drop (Fig. 100), then by drawing tangents from _O_ (the point of mutual contact), {296} we determine the three angles of our triangle (Fig. 101), and we therefore know the relative magnitudes of the three surface tensions, which magnitudes are proportional to its sides; and conversely, if we know the magnitudes, or relative magnitudes, of the three sides of the triangle, we also know its angles, and these determine the form of the section of the drop. It is scarcely necessary to mention that, since all points on the edge of the drop are under similar conditions, one with another, the form of the drop, as we look down upon it from above, must be circular, and the whole drop must be a solid of revolution. ―――――――――― The principle of the Triangle of Forces is expanded, as follows, by an old seventeenth-century theorem, called Lami’s Theorem: “_If three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle contained between the directions of the other two._” That is to say _P_ : _Q_ : _R_ : = sin _QOR_ : sin _POR_ : sin _POQ_. or _P_/sin _QOR_ = _Q_/sin _ROP_ = _R_/sin _POQ_. And from this, in turn, we derive the equivalent formulae, by which each force is expressed in terms of the other two, and of the angle between them: _P_^2 = _Q_^2 + _R_^2 + 2_Q_ _R_ cos(_QOR_), etc. From this and the foregoing, we learn the following important and useful deductions: (1) The three forces can only be in equilibrium when any one of them is less than the sum of the other two: for otherwise, the triangle is impossible. Now in the case of a drop of olive-oil upon a clean water surface, the relative magnitudes of the three tensions (at 15° C.) have been determined as follows: Water-air surface 75 Oil-air surface 32 Oil-water surface 21 No triangle having sides of these relative magnitudes is possible; and no such drop therefore can remain in equilibrium. {297}