NOTICES AND ABSTRACTS 



MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 



MATHEMATICS AND PHYSICS. 



Mathematics. 



On the Polyhedron of Forces. By J. T, Graves, M.A., F.R.S. 



If any number of forces, represented in number and magnitude by the faces of a 

 polyhedron, and in direction perpendicular to those faces, act upon a point, they 

 will keep it in equilibrium. The above is the proposition which is called by the 

 writer " the Polyhedron of Forces." It has probably occurred to many, that the 

 well-known geometrical representation in magnitude and direction of a system of 

 balanced forces acting upon a point by the sides of a closed polygon is so simple 

 and complete that nothing needs to be noted beyond the polygon of forces. What 

 is commonly called the parallelopipedon of forces — which is the elementary theorem 

 in solid space analogous to the parallelogram of forces — represents by the" diagonal 

 of a parallelopipedon the resultant force, which balances the three forces represented 

 by the areas. But there the separate forces are represented by lines. The writer 

 was led more than ten years ago to the representation of forces by areas in making 

 researches respecting complex numbers with a new imaginary symbol. He has 

 mentioned the result here enunciated to several mathematicians, to whom it has 

 appeared familiar, and who have believed that it must have been already published ; 

 but the writer has searched for it in collections of memoirs and works on statics, 

 and has been unable to find it in print. He has, accordingly, been advised by a 

 very learned scientific friend to occupy it, if it has not been already appropriated. 

 With this view, he takes this opportunity of publishing it to the British Association. 



On the Congruence nx^=^n-\-\ (mod/)). By John T. Graves, M.A., F.R.S. 



As is well known to those who have studied foreign works on the theory of num- 

 bers, the expression 



a =: b (mod. c) 

 denotes that a — 6 divided by c is a whole number. When this relation has place, 

 a and 6 are said to be congruent with respect to the modulus c, and the relation 

 itself is called a congruence. 



Mr. J. T. Graves shows, from elementary principles of the theory of numbers, 

 that in the congruence 



«ar = n-t-l (mod. ^), 

 if p be a prime number, and if m be made to assume, in regular ascending order, all 

 values from 1 top— 1 inclusive, x will be found to have, in some order or other, all 

 values from 2 to p inclusive. 



1856. 1 



