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XXII. — On the Decomposition of Forces externally applied to an Elastic Solid. 

 By W. J. Macquorn Rankine, C.E., LL.D., F.K.SS. L. & E. 



(Received, 5th January ; read, 15th January, 1872.) 



Introductory Remarks. — The principles set forth in this paper, though now 

 (with the exception of the first theorem) published for the first time, were com- 

 municated to the French Academy of Sciences fifteen years ago, in a memoir 

 entitled " De l'Equilibre int^rieur d'un Corps solide, elastique, et homogene," 

 and marked with the motto, " Obvia conspicimus, nubem pellente Mathesi," 

 the receipt of which is acknowledged in the Comptes Rendus of the 6th April 

 1857. 



(1.) Principle of Isorrhopic Axes. — The following theorem was first pub- 

 lished in the " Philosophical Magazine" for December 1855. 



Prop. I. " Theorem. Every self-balanced system of forces applied to a con- 

 nected system of points is capable of resolution into three rectangular systems 

 of parallel self-balanced forces applied to the same points. 



" Demonstration. — Assume any set of rectangular axes, to which reduce the 

 forces and the positions of their points of application ; and let X, Y, Z be the 

 components of the force applied to any point (x y z)" 



"Let 



?.Xx = A;2.Yy = B;?.Zz = C;'2.Yx = 2,.Zy = D;2.Zx 



= y 2.Xx = E;I,.Xy = '2.Yx = F.- 



Then, in linear transformations of rectangular co-ordinates, A is covariant with 

 x 2 , D with yz, &c. 



" Conceive the surface of the second order, whose equation is 



Ax 2 + By 2 + Cz 2 + 2Byz + 2Ezx + 2Fxy = constant . (1). 



Then if the forces, and the positions of their points of application be reduced 

 anew to the principal axes of that surface, we shall have 



D = 0, E = 0, F = ; 



and consequently, each of the three systems of component forces parallel to 



VOL. XXVI. PART IV. 8 Z 



