292 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



in the second case, each pair of the triangles a/3<T', /3y<5', ya<T will 

 overlap, at least when the tensions cr DA , <T DB , O"DO are on ly a little too 

 great to be represented as in figure 15, and the sum of the angles of 

 each of the triangles adb, bdc, cda will be greater than two right 

 angles. 



Let us denote by i> A , V E , V G the portions of V D which were originally 

 occupied by the masses A, B, C, respectively, by S DA , S DB , S DC , the 

 areas of the surfaces specified per unit of length of the mass D, 

 and by S AB , S BO , S OA , the areas of the surfaces specified which were 

 replaced by the mass D per unit of its length. In numerical value, 

 ^A> v v> v c w iH b e equal to the areas of the curvilinear triangles 

 bed, cad, abd', and S DA , S DB> S DC > S AB , S BC , S CA to the lengths of the 

 lines be, ca, ab, cd, ad, bd. Also let 



^s = "DA SDA + "DB SDB + cr DC s D c <j AB S AB cr BC S BC cr CA S CA , (626) 

 and Wv=pv I >-p A yi.-pxV B -p G v . (627) 



The general condition of mechanical equilibrium for a system of 

 homogeneous masses not influenced by gravity, when the exterior 

 of the whole system is fixed, may be written 



2(<r&)-Z(patO0. (628) 



(See (606).) If we apply this both to the original system consisting 

 of the masses A, B, and C, and to the system modified by the 

 introduction of the mass D, and take the difference of the results, 

 supposing the deformation of the system to be the same in each 

 case, we shall have 



O"DA ^DA "I" tf'DB <^DB H~ "DC O^DC <T AB OS AB <T B o OS B 



- <7 CA &OA -Pi> &>D +PA Sv A +p E Sv B +p G 8v c = 0. (629) 

 In view of this relation, if we differentiate (626) and (627) regarding 

 all quantities except the pressures as variable, we obtain 



d W s d W y = S DA do- DA + SDB ^DB + s D c ^DO 



S AB <^o- AB S BC ^O"BC S CA ^O"CA (630) 



Let us now suppose the system to vary in size, remaining always 



similar to itself in form, and that the tensions diminish in the 

 same ratio as lines, while the pressures remain constant. Such 



changes will evidently not impair the equilibrium. Since all the 

 quantities S DA , o- DA , S DB , <r DB , etc. vary in the same ratio, 



SDA^DA^^DASDA), s DB do- DB = Jd(<r DB s DB ), etc. (631) 

 We have therefore by integration of (630) 



TT 8 "^v = i ("DA SDA + <T DB S DB + O- DO S DC O*AB SAB CT BO S BO ^OASCA)* (632) 



whence, by (626), 



(633) 



