RECIPROCAL FIGURES, FRAMES, 



Here the function / which determines the stress in the strata parallel to xy is 



Now, thia function is not sufficiently general, for instead of being any 

 function of x, y, and z, it is the product of a function of x and y multiplied 

 by a function of z. 



Besides this, though the value of p a is, as it ought to be, a function of x 

 and y only, it is not of the most general form, for it depends on G, t he- 

 function which determines the stresses p a , p^, and p n , whereas the value of 

 p m may be entirely independent of the values of these stresses. In fact, the 

 equations give 



This method, therefore, of representing stress in a body of three dimen- 

 sions is a restricted solution of the equations of equilibrium. 



On Reciprocal Diagrams in Three Dimensions. 



Let us consider figures in two portions of space, which we shall call respec- 

 tively the first and the second diagrams. Let the co-ordinates of any point in 

 the first diagram be denoted by x, y, z, and those of the corresponding point in 

 the second by i), > measured in directions parallel to x, y, z respectively. 

 Let F be a quantity varying from point to point of the first figure in any 

 continuous manner ; that is to say, if A, B are two points, and F u F t the 

 values of F at those points ; then, if B approaches A without limit, the value 

 of F, approaches that of F t without limit. Let the co-ordinates (f, rj, ) of a 

 point in the second diagram be determined from x, y, z, those of the corre- 

 sponding point in the first by the equations 



, dF dF dF 



This is equivalent to the statement, that the vector (p) of any point in the 

 second diagram represents in direction and magnitude the rate of variation of /' 

 at the corresponding point of the first diagram. 



