716 PROFESSOR MACQUORN RANKINE ON THE DECOMPOSITION OF 



those three principal axes will be self-balanced, independently of the other two 

 systems. Q.E.D"* 



Remark.— The values of A, B, &c, obviously depend solely on the directions 

 of the axes, and not on their point of intersection. 



(2.) Definitions. — The following terms will be employed in the sequel, rela- 

 tively to any given system of forces. 



Rhopimetric Surface. — The surface (1). 



Rhopimetric Co-efficients. — The quantities A, B, C, D, E, F. 



Isorrhopic Axes. — The principal axes of the surface (1). 



Principal Rhopimetric Co-efficients. — The values of the co-efficients A, B, C, 

 for the isorrhopic axes. 



Arrhopic System.— A system of forces for which A = 0, B = 0, C = 0, 

 D = 0, E = 0, F = 0; and for which, consequently, every direction is an isor- 

 rhopic axis. 



(3.) Application to Elastic Solids. — The utility of the above principle of isor- 

 rhopic axes in the theory of the equilibrium of elastic solids arises from the 

 fact, that although, in treating of the equilibrium of a solid body as a whole 

 supposed to be perfectly rigid, it is allowable to suppose the point of appli- 

 cation of any force to be anywhere in the line of action of that force ; yet, when 

 the solid body is considered as being strained by the forces applied to it, no 

 such supposition is admissible ; and in every mathematical process for deter- 

 mining such straining effect the actual point of application of each force must 

 alone be considered. When the straining forces to which an elastic solid is 

 subjected are restricted within certain limits, the straining effect of any number 

 of self-balanced systems of forces combined is sensibly equal to the sum of the 

 effects which those systems respectively produce when acting separately. 



Consequently, the principle of Isorrhopic Axes affords the means of re- 

 ducing the problem of finding the straining effect of any self-balanced system 

 of forces applied to an elastic solid to that of finding the separate straining 

 effects of three self-balanced systems of parallel forces. 



Prop. II. — Problem. To find the Rhopimetric Co-efficients for a system of 

 Forces applied over the surface and throughout the interior of a solid body. 



Let X, Y, Z, denote the components of the attractive or repulsive accelera- 

 tive force applied to a molecule of the solid whose co-ordinates are x, y, z, its 

 volume dx dy dz, and its density p ; let P, Q, R, denote the components of 

 the external stress (tension positive) per unit, of area, which acts on an element 



* It is easy to see how this theorem may be extended, to a system of moving masses, by putting 



X — m—jTf for X, &c 



