RECIPROCAL DIAGRAMS IN SPACE. 103 



molecular forces in solid bodies. I shall indicate two independent methods of 

 representing internal stress by means of reciprocal diagrams. 



First Method. Let a, b be any two contiguous points in the first diagram, 

 and a, y8 the corresponding points in the second. Let an element of area be 

 described about ab perpendicular to it. Then if the stress per unit of area on 



this surface is compounded of a tension parallel to a/8 and equal to P~ and 



CltV 



11 7 D , 



a, pressure parallel to cut and equal to P ( +~r^ + -r , P being a constant 



introduced for the sake of homogeneity, then a state of internal stress denned 

 in this way will keep every point of the first figure in equilibrium. 



The components of stress, as thus defined, will be 



p p 



~ ' Pzx ~ ' Pxy ~ 



dydz' zx ~ dzdx ' xy ~ dxdy ' 



and these are easily shewn to fulfil the conditions of equilibrium. 



If any number of states of stress can be represented in this way, they 

 can be combined by adding the values of their functions (F), since the quan- 

 tities are linear. This method, however, is applicable only to certain states of 

 stress ; but if we write 



_d 3 B &C _<fC #A _d\A d?B 



Pzx ~ dz* + dy 3 ' Pvy ~dx* + dz* ' Pzz ~ dif + dx* ' 



(PA d"B &C 



Pyz ~ dydz' Pzx ~ dzdx' Pxy ~ dxdy' 



we get a general method at the expense of using three functions A, B, C, 

 and of giving up the diagram of stress. 



Second Method. Let a be any element of area in the first diagram, and 

 a the corresponding area in the second. Let a uniform normal pressure equal 

 to P per unit of area act on the area a, and let a force equal and parallel 

 to the resultant of this pressure act on the area a, then a state of internal 

 stress in the first figure defined in this way will keep every point of it in 

 equilibrium. 



