718 



PROFESSOR MACQUORN RANKINE ON THE DECOMPOSITION OF 



T x being considered as positive when it tends to elongate the positive and 

 shorten the negative diagonal of the faces ±x, &c. In the transformation of 

 co-ordinates, N x , N^ N 2 , T^., T , T s , are covariant respectively with x 2 , y 2 , z 2 , yz, 



zx, xy. Let n x , n y , n z , be the direction-cosines of the external normal to any 



point of the body's surface. 



Then the following are the conditions of equilibrium between the internal 

 stresses and the applied forces. 



Conditions relative to each Internal Molecule : — 



dx 



dT z 



dx 



dTy 

 dx 





+ 



dy 



dN y 

 dy 



dT r 



dz 

 dT 



dy 



+ 



dz. 



dz 



+ P Z = 



/ 



(3). 



Conditions relative to each Point of the Surface : — 



P = n x ~N x + n y T z + n z T y 

 Q = n x % + w,N y + nJT m 

 E = n x T y + n y T x + n z ~N z 



(4). 



(5.) Effect of Terrestrial Gravitation. — In a homogeneous heavy body near 

 the earth's surface, the internally applied forces pX, pY, pZ, are constants, 

 being simply the components of the weight of unity of volume of the body 

 along the three axes of co-ordinates. 



Prop. III. — Problem. To Balance the Weight of a Homogenous Body by 

 Pressure applied to its Surface, so as to form an Arrhopic System of Forces ; and 

 to Determine the corresponding Internal Stresses. — Assume a vertical direction 

 positive downwards, for the axis x, and let the plane of yz pass through the 

 centre of gravity of the body; then pY = 0, pZ = ; and pX = gp is the weight 

 of unity of volume of the body. For the applied external pressures, make 

 Q, = 0, ft, = 0, 



P g = — gpx'n x ; 



(5). 



Then the system of pressures P ? balances the weight of the body ; for let the 

 horizontal projection of an element d 2 s of the body's surface be dy dz, then d 2 s = 



d/u dz 



-f: — ; and that projection is to be considered as positive or negative according 



x 



to the sign of n x ; consequently 



ffVd's = - gpffxfdydz = - g P Y. 



