28 



ME CLERK MAXWELL ON 



manifest when we deduce from it the diagram of stress, the co-ordinates of whose 

 points are 



t = -j- and « = 



dx 



(21). 



For if s be the length of any curve in the original figure, and o- that of the cor- 

 responding curve in the diagram of stress, and if Xds, Yds are the components 

 of the whole stress acting on the element ds towards the right hand of the 

 curve s 



Xds _ „ ( Jy f F cJ i fh -. d Z c !i (h _ <*£ ll(T 



j\xlo — J/xx ~tT Llo — ~~ 7 9 ^^ Clo — -5— ^^ (to — —j— U (J 



and 



dy 2 ds 



,, , dx , c/ 2 F cfo 7 



Y* = - Ar 3T-& =~ si t. ds = ffi 5 & = J & 



<7y da; 1 _dy 



(22). 



VZs 



rfa; 2 cZs 



Hence the stress on the right hand side of the element ds of the original curve 

 is represented, both in direction and magnitude, by the corresponding element 

 da of the curve in the diagram of stress, and, by composition, the resultant 

 stress on any finite arc of the first curve s is represented in direction and 

 magnitude by the straight line drawn from the beginning to the end of the 

 corresponding curve o\ 



If Pj, P 2 are the principal stresses at any point, and if P x is inclined a to the 

 axis of x, then the component stresses are 



p xx = Pj cos 2 a + P 2 sin 2 a ) 



p xy = (P x — P 2 ) sin a cos a I . . (23). 



p yy = T 1 sin 2 a + P 2 cos 2 a J 



Hence 



tan 2a = 



P 1 + P 2 



P*> — 



Pxx Pyy 



Pxx "T ]?yy '■ 



d 2 F 

 dxdy 



d*F 



-T i-E 2 — Pxx Pyy Pxy — 



d 2 F d 2 F 

 dx 3 dy 2 



cPFd?F 

 dx 2 dy 2 



cPF 

 df 



dxdy 



(24). 



) 



Consider the area bounded by a closed curve s, and let us determine the sur- 

 face integral of the sum of the principal stresses over the area within the curve. 

 The integral is 



By a well-known theorem, corresponding in two dimensions to that of Green in 

 three dimensions, the latter expression becomes, when once integrated, 



f(dF dx dF dy\ 

 J \dy ds dx ds) 



(26), 



