FORCES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 719 



where V is the entire volume of the body. Also the entire system of forces is 

 Arrhopic. For it is evident that 



B = C = D = E = F = 0; 

 and 



A = gp {fffx dx dy dx —ffx' 2 dy dz) . 



Now the first term of the above expression is well known to be null when the 

 plane yz traverses the centre of gravity ; and by attending to the rule, that 

 dydz is positive or negative according to the sign of n x , it appears that 

 the second term is null also. Therefore, A = 0, and the system of forces is 

 Arrhopic. 



Lastly, for the system of internal stresses, make 



N f . f = 0; N 2 ., = 0; T,., = 0; T y . g = ; %. g = ; N..,= - gpx . (6). 



This system evidently satisfies equations 3, 4, and 5 ; that is to say, the re- 

 quired system of internal stresses consists in a vertical normal stress at each 

 molecule, proportional to its vertical distance from the horizontal plane of the 

 centre of gravity of the body, tensile above that plane, and compressive below. 

 Q.E.I. 



Definitions. — Antibarytic Pressures: the externally applied pressures 

 which (as in the above problem) form, with the gravitation of a body, an 

 Arrhopic system. Antibarytic Stresses : the corresponding internal stresses. 



Remark. — It is characteristic of the Antibarytic pressures that their inten- 

 sity for each unit of area of the horizontal projection of the body's surface is a 

 linear function of the vertical co-ordinate, viz., 



& = - gpaf ...... (6A). 



n x 



Abarytic Pressures.— The system of pressures left after taking away the 

 Antibarytic Pressures from the Actual Pressures applied at the several elements 

 of the body's surface. 



Corollary. — The Abarytic Pressures are self-balanced ; their Ehopimetric 

 co-efficients are the same with those of the] entire system of Applied Forces ; 

 and in calculating their effects, the force of gravity is to be left out of consider- 

 ation. Hence the internal stresses corresponding to a system of Abarytic 

 Pressures must fulfil the following equations : — 



H** + ^L ,_^ _ • 

 dx dy dz ~ ' 



dX dN^ ,dTz_ 

 dx "*" dy + dz ~ U ' 



dT v dT x d~N z 



dx ^ dy dz ' 



VOL. XXVI. PART IV. 9 A 



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