TPIE FIGURES OF EQUILIBRIUM OF A LIQUID MASS 269 



ample equally curious. I have given, in the second series, as the outline of 

 that system, the one which I reproduce here (Fif?. 21,) i;i 

 which the liquid edge a h is common to the four iilms which 

 proceed from the iour oblique edges of the frame-work ; but 

 what then deceived me was that, with oil and in the interior 

 of the alcoholic liquid, this edge preserves so considerable a 

 thickness that it is impossible to raise it without occasioning a 

 rupture of one of the films — a thickness which maintains the 

 stability of the system; now Avhen we realize the laminar sys- 

 tem of this frame by means of the glyceric liquid, the edge in 

 question is found to be replaced by an additional lamina, (Fig. 22,) and then 

 each liquid edge is common to no more than three films. 



The pyramid of figures 21 and 22 has more height than that 

 Avhich is represented in the second series ; it is because wi:h this 

 last the laminar system obtained by means of the glyceric liquid 

 always presents, from its formation or shortly after, a slight ir- 

 regularity in the position of the four liquid edges directed to- 

 wards the summits of the base — an irregularity which is not pro- 

 duced with a higher pyramid. I shall recur in the sequel to this 

 irregularity, which connects itself Avith an order of facts here- 

 after to be examined, and of which I shall then indicate the 

 cause. 



§ 17. Let us at present remark that in the systems of figures 15 and 19 there 

 are four edges terminating at the same point, namely, the three which unite two 

 by two the spherical caps and that which unites the three ])artitions. Now, as 

 has been seen, (oth series, § 19,) the same fact presents itself in all our laminar 

 systems obtained with the glyceric liquid : the liquid edges which terminate at 

 the same liquid point are always four in number; here again, then, there is a 

 general law of laminar assemblages. It may be added that if we apply to the 

 point in question the considerations advanced (§ S) in regard to edges, we shall 

 conclude that these must intersect each other at this point under angles either 

 equal or exceedingly near equality. And, in fact, the junction of the small 

 concave surfaces pertaining to the edges produce, necessarily, around the point 

 where they unite, four small surfaces strongly concave in all directions — sur- 

 faces whose capillary equilibrium requires equal or very nearly equal, curva- 

 tures!, which implies the absolute equality, or very nearly such, of the angles 

 which we are considering. 



Let us seek the value of these angles, supposing their equality to be absolutely 

 exact. When the liquid edges are curved, it is proper to replace the arcs by 

 their tangents at the point where they terminate, so that it suffices, in all cases, 

 to imagine four right lines terminating at a single point under equal angles. In 

 order, therefore, to follow the most simple method, let us consider a regular te- 

 Fiff. 23 trahedron abed, (Fig. 23.) Let o be the centre of this te- 



trahedron; draw the lines oa, ob, oc, od to the four summits; 

 these lines will evidently form with one another equal angles, 

 whose value will consequently be that which it is proposed to 

 obtain. This being premised, let us prolong the line ao to 2^> 

 where it attains the base, and join ^^(Z; the triangle o p d will 

 be rectangular at p. Let it be remarked now that the point o is 

 the centre of gravity of the tetrahedron, and that the point p is the centre of 

 gravity of the base b cd; now we know that if, in any pyramid, we join the summit 

 to the centre of gravity of the base, the centre of gravity of the pyramid is situated 

 on that right line, at three-quarters of its length reckoning from the summit; 

 op is, therefore, the third of oa, and as the point o is at an equal distance from 

 the four summits, op is also the third of od. In the rectangular triangle o p d 

 we have consequently cos doj) = ^ ; whence results cos doa = — ^. Thus, 



