468 THE FORMS OF TISSUES [ch. 



magnitudes are proportional to the specific tensions characteristic 

 of the three " interfacial " surfaces. Now for three forces acting at 

 a point to be in equihbrium they must be capable of representation, 

 in magnitude and direction, by the three sides of a triangle taken 

 in order, in accordance with the theorem of the Triangle of Forces. 

 So, if we know the form of our drop as it floats on the surface 

 (Fig. 152), then by drawing tangents P, R, from (the point of 

 mutual contact), we determine the three angles of our triangle, and 

 know therefore the relative magnitudes of the three surface-tensions 

 proportional to its sides. Conversely, if we know the three tensions 

 acting in the directions P, R, S (viz. Tab, T^c, T^^) we know the three 

 sides of the triangle, and know from its three angles the form of 

 the section of the drop. All points round the edge of the drop being 

 under similar conditions, the drop must be circular and its figure 

 that of a sohd of revolution*. 



The principle of the triangle of forces is expanded, as follows, 

 in an old seventeenth-century theorem, called Lamy's Theorem : 



// 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 (in Fig. 152) 



P : R : S = sin (f) : sin p : sin s, . 



P R S 



or ^-7 = -^ — ==-■ — • 



sm (f) sm p sm s" 



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: viz. 



P^ = R^ + S^+ 2RS cos </., etc. 



From this and the foregoing, we learn the following important 

 and useful deductions: 



(1) The three forces can only be in equihbrium when each is less 



* Bubbles have many beautiful properties besides the more obvious ones. For 

 instance, a floating bubble is always part of a sphere, but never more than a 

 hemisphere; in fact it is always rather less, and a very small bubble is considerably 

 less, than a hemisphere. Again, as we blow up a bubble, its thickness varies 

 inversely as the square of its diameter; the bubble becomes a hundred and fifty 

 times thinner as it grows from an inch in diameter to a foot. In an actual calculation 

 we must always take account of the tensions on both surfaces of each film or 

 membrane. 



