VII] OF CLUSTERED BUBBLES 485 



plane drawn through their centres, c, c', c" (or what is the same 

 thing, if it represent the base of three bubbles resting on a plane), 

 then the lines uc, uc", or sc, sc', etc., drawn to the centres from the 

 points of intersection of the circular arcs, will always enclose an 

 angle of 60°. Again (Fig. 171), if we make the angle c"uf equal 

 to 60°, and produce uf to meet cc" in/, / will be the centre of the 

 circular arc which constitutes the partition Ou; and further, the 

 three points /, g, h, successively determined in this manner, will lie 

 on one and the same straight line. In the case of three co-equal 

 bubbles (as in Fig. 170, B), it is obvious that the hnes joining their 

 centres form an equilateral triangle: and consequently, that the 

 centre of each circle (or sphere) lies on the circumference of the 

 other two; it is also obvious that uf is now parallel to cc'\ and 

 accordingly that the centre of curvature of the partition is now 

 infinitely distant, or (as we have already said) that the partition 

 itself is plane. 



The mathematician will find a more elegant way of dealing with our 

 spherical bubbles and their associated interfaces by the method of spherical 

 inversion, (i) Take three planes through a line, cutting one another at 60°, 

 and invert from any point, arid you have the case of two spherical bubbles 

 fused, with their interface also spherical, (ii) Take the six planes projecting 

 the edges of a regular tetrahedron from its centre, and you get by inversion 

 the case of the three unequal bubbles and their three interfaces, (iii) Take 

 these same planes with a bubble added centrally (thus adding a spherical 

 tetrahedron), and inversion gives the general case of four fused bubbles and 

 their six spherical partitions. 



When we have four bubbles meeting in a plane (Fig. 172), they 

 would seem capable of arrangement in two symmetrical ways: 

 either (a) with four partition-walls intersecting at right angles, 

 or (6) with five partitions meeting, three and three, at angles of 120°. 

 The latter arrangement is strictly analogous to the arrangement of 

 three bubbles in Fig. 170. Now, though both of these figures might 

 seem, from their apparent symmetry, to be figures of equilibrium, 

 yet in point of fact the latter turns out to be of stable and the 

 former of unstable equilibrium. If we try to bring four bubbles 

 into the form (a), that arrangement endures only for an instant; 

 the partitions glide upon one another, an intermediate wall springs 

 into existence, and the system assumes the form (6), with its two 



