[ 92 ] 



XVII. Princijj/c of the Equilibrium of Polyhedral Frames. 

 By W. J. MAcquorn Rankine, C.E., LL.D., FR.SS.L.fyE* 



^|^IIE following theorem is the extension to polyhedral frames 

 J- of a principle which is proved for polygonal frames in ' A 

 Manual of Applied Mechanics/ art. 150. 



Theorem. — If planes diverging from a point or line be drawn 

 normal to the lines of resistance of the bars of a polyhedral frame, 

 then the faces of a polyhedron whose edges lie in those diverging 

 planes (in such a manner that those faces, together with the diver- 

 ging planes which contain their edges, form a set of contiguous 

 diverging pyramids or wedges) will represent, and be normal to, 

 a system of forces which, being applied to the summits of the 

 polyhedral frame, will balance each other — each such force being 

 applied to the summit of meeting of the bars whose lines of 

 resistance are normal to the set of diverging planes that enclose 

 that face of the polyhedron of forces which represents and is 

 normal to the force in question. Also, the areas of the diverging 

 planes will represent the stresses along the bars to whose lines 

 of resistance they are respectively normal. 



It is obvious that the polyhedron of forces and the polyhedral 

 frame are reciprocally related as follows : their numbers of edges 

 are equal, and their corresponding pairs of edges perpendicular 

 to each other ; and the number of faces in each polyhedron is 

 equal to the number of summits in the other. 



Glasgow, January 9, 1864. 



XVIII. On the Theory of the Velocity of Sound. 

 By Professor Challis, M.A., F.R.S., F.R.A.S* 



THE « Note " of Professor Tyndall " On the Velocity of 

 Sound " in the Number of the Philosophical Magazine for 

 last November, and the reference therein made to Dulong's expe- 

 riments, have led me to see that the principles I have applied to 

 this question admit of an extension which had not previously 

 occurred to me. For this reason, and because I am desirous of 

 making a few remarks on the notice taken of my researches by 

 Prof. Le Conte in the article inserted in the January Number, I 

 now revert to the subject. 



The theoretical value of the velocity of sound which I have 

 obtained, agreeing very closely with the observed value, is a 

 purely mathematical deduction, on hydrodynamical principles, 

 from the hypotheses that the medium is a perfect fluid, and that 



* Communicated by the Author. 



