272 WITHDRAWN FEOM THE ACTION OF GEAVITY. 



terminating at two opposite summits of the frame, the planes of the two quadri- 

 laterals are at a right angle one with the other. 2. The twelve films proceeding 

 from the solid edges, films each of which is triangular and limited at two of the 

 long edges of the above quadrilaterals. There is in each of these quadrilaterals 

 a dimension easy to measure by the cathetometer : it is the distance comprised 

 between the summits t and u of the two opposite obtuse angles. Recurring to 

 the principle that these two angles, as well as that whose summit is at the cen- 

 tre of the figure, must have — J as cosine, it is easily demonstrable that this 

 distance t u must be exactly the third of the length of the edges of the octahedron. 

 To show this, let stou (Fig. 27) be one of the quadrilaterals; draw Rg. 27 

 the two diagonals tu and os, which shall intersect at a right angle 

 at m; let the distance iu be designated by d, the common value of 

 the three obtuse angles by a, and the length of the edge of the oc- 

 tahedron by a. In the triangle ots, the angle at t being a, and the 

 angle at o being ^a, the angle at s will be 180° — -|a; according 

 to this, the two rectangular triangles omt and smt will give om = 



-,3i\\(\.ms = TT^TT^ — 5— ;• Let us remark now that the 



tangja tang(180° — |a) 



diagonal os, or the sum of the two lengths om and 7ns, is half the height of the 



octahedron, and consequently, as will be readily seen, is equal to —j-^. We shall 



have then -J^ I ;— + -— — — ^ — \ I = —7- But knowing that the co- 



"^ Ltang^a tang(180° — f«/J y2. 



sine of the angle a is — ^, we shall find tang^a = -\/2, and tang(180° — |a) == 



i-v/S. These values being substituted in the above formula, we obtain, as 



before said, d = —. In my frame, the measurements gave* for a the value 



g9mm_49^ of which the third is 23"i™.16; and for the distance d, the value 

 23"^°^.14. The difference 0"^'^.02 between the value calculated and the value 

 measured, it will be seen, is even more insignificant than in the preceding case. 



I have said (5th series, § 19) that a solid frame being given as to its form, 

 it might be proposed, as a geometric problem, to occupy the interior with an 

 assemblage of surfaces subjected to the laws which govern my laminar systems ; 

 now this has been done by M. Lamarle in regard to the regular octahedron by 

 supposing all the surfaces to be plane, and he has thus found a system identical 

 with that which he had seen produced in my frame. 



§ 20. The laminar systems of other frames — those, namely, which contain 

 curved films, and consequently curved liquid edges — also verify, and in a very 

 curious though somewhat less precise manner, the equality of the angles under 

 which these edges terminate at the same liquid point ; they also verify, at least 

 in the immediate vicinity of the latter, (§ 17,) the equality of the angles between 

 the three films which join one another at each of these same edges. 



* A regular octahedral frame may be considered as formed of an assemblage of three 

 squares of iron wire whose planes cut one another diagonally, whence the frame was so 

 placed that these squares should present themselves successively in front of the cathetometer 

 with two of cheir sides directed vertically, and, in the three positions of the frame, the dis- 

 tance comprised between the two horizontal wires was measured as near as possible to each 

 of these vertical wires. Thus were obtained twelve values, whose mean was 68™'^.59 ; 

 moreover, eight measurements of the diameter of different wires were taken, which gave as a 

 mean O^^.OU. For the reasons stated in the preceding note, this diameter was added to the 

 former number, and thus was found the value of a indicated in the text. As regards that of 

 d, since small irregularities of the frame might introduce slight differences between the six 

 quadrilaterals, that distance was measured in each of them, and the value of </ given in the 

 text is the mean of those six measui'ements. 



