578 REPORT— 1894. 



2. On Ronayne's Cubes. By Professor H. Hennessy, F.R.S. 



Some years since a box containing a pair of equal cubes was placed in the 

 author's hands, and he found that one of these could have its parts displaced so as 

 to leave a peculiarly shaped shell, through which the second cube passed without 

 any difficulty. Groups of twin crystals of a cubical form have been long known, 

 but their grouping could rarely admit of such a structure as the cubes referred to 

 present. 



It is manifest that a cube passed through another in the direction of the 

 diagonal of the square would leave two triangular prisms, but in order to connect 

 these prisms two flanges with interior sloping faces should be attached. The 

 thickness of these flanges in two directions, as well as the angle of inclination of 

 the sloping faces, are all connected by geometrical conditions which permit of the 

 solution of the problem of the construction of the shell of the first cube. The 

 author was unable to find the solution of the problem originally published by the 

 inventor of the cubes, Mr. J. Ronayue, some time about the middle of the last 

 century. Under these circumstances he completed the inquiry, and on comparing 

 the results with the measured dimensions of the prisms and flanges which con- 

 stitute the shell of the first cube he found the perfect concordance between the 

 calculated and measured dimensions. "When x represents the distance from a 

 corner of the cube to the edge of the flange, and a the side of the cube, then 6, the 

 inclination of the sloping face to the face of the cube, is found to be represented by 



sm 6 = i — '— 



3«5 + 4,z^-4a.iV2 



The edge of each cube is 1'92 inch, and 6 is found both by measurement and by 

 the above formula to be 9° 45' nearly. 



3. On a Pro}oerty of the Catenary. By Professor H. Hennessy, F.R.S. 



In the course of inquiry into some hydraulic questions the author found that 

 the catenary of maximum area under a given perimeter may be inscribed in a 

 semicircle.^ Hence, if the radius of the semicircle is one foot, the chain hung 

 within it when in a vertical plane will be one yard. Thus the two fundamental 

 standards of English measure are connected with the catenary of maximum area. 



4. A Complete Solution of the Problem, ' To find a Conic with respect to 

 which two given Conies shall be Reciprocal Polars.' By J. W. Russell, 

 M.A. 



In the author's ' Elementary Treatise on Pure Geometry,' p. 147, a construc- 

 tion is given in the case in which the given conies intersect in distinct points. 

 This construction was extended to the cases of the conies touching or having three- 

 point contact. The method in the case of double contact was dirt'ereut. Taking 

 U to be the common pole and AB the common chord, through U draw any line 

 meeting the conies in PP', QQ' respectively. Let X Y be the double points of 

 the volution PQ, P'Q' or of PQ', P'Q. Then the conic required is the conic 

 touching UA at A and UB at B, and passing through X or Y. This method holds, 

 when suitably modified, in the case of four-point contact. 



5. The Impossibility of Trigraphic Fields of Spaces. 

 By J. W. RussELi, JI.A. 



The simplest trigraphic form is that generated by three points P, P', V" on 

 three lines AB, A'B', A"B'', these points being connected by a relation of the 

 form 



a^k' k' ¥ + a^k' k" + ... + ajc + ... -i- Og = 



' Proc. Roy. Soc, vol. xliv. p. 106. 



