FORCES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 717 



of the external surface of the solid whose co-ordinates are x', tf, z, and area 



d 2 s; then 



A =fffxX. P . dx dy dz + ffx'Fd 2 s ; \ 

 D =fffzY P . dx dy dz + ffz'Q . d 2 s I . . (2), 



=fffyZp . dx dy dz + ffy'K . d 2 s ) 



and the expressions for the other co-efficients will be similar, mutatis mutandis. 

 Q.E.I. 



In the case of normal external stress, let n x , n y , n z be the direction-cosines 

 of the normal to the element d 2 s, and S the intensity of normal stress on 

 that element ; then 



P = Sra, ; Q = Sn v ; E = Sn, ; 



and in finding the values of the double integrals, we may put n x d 2 s = dy dz, 

 &c ; observing, that for each set of an even number of elements d 2 s, which 

 have a common projection such as dy dz, the quantity to be integrated is of a 

 form such as 2S#', and contains as many terms as there are elements having a 

 common projection ; the sign of each term being positive or negative, accord- 

 ing as the direction-cosine (as %) is positive or negative. 



(4.) Summary of the Relations between Internal Stresses and Applied Forces 

 in an Elastic Solid in Equilibrio. 



The following principles having been long known through the investigations 

 of various mathematicians, are here recapitulated for the sake of convenience. 

 In expressing internal stresses, the notation of M. Lame is adopted, viz., let 

 dx dy dz be a rectangular molecule, and let 



N ,N ,N , 



be the normal stresses per unit of area, on the pairs of faces normal respec- 

 tively to 



x, y, *, 



such stresses being considered as positive when tensile, negative when compres- 

 sive. Also, let 



T T T 



be the tangential stresses per unit of area on the double pairs offices parallel 

 respectively to 



oo, y, z. 



