[Proceedings of the Aonrfon Mathematical Society, Vol. II.] 



XXXIII. On Reciprocal Diagrams in Space, and their relation to 



Function of Stiress. 



LET F be any function of the co-ordinates x, y, z of a point in space, and 

 l't *? be another system of co-ordinates which we may suppose referred 

 t> axes parallel to the axes of x, y, z, but at such a distance (in thought) 

 that figures referred to z, y, z do not interfere with the figures referred to f, 17, . 



We shall call the figure or figures referred to x, y, z the First Diagram, 

 and those referred to f, 77, the Second Diagram. 



Let the connection between the two diagrams be expressed thus 



(IF _dF (IF 



*~dx' V'lly' ^~dz' 



When the form of F is known, rj and may be found for every value 

 of x, y, z, and the form of the second diagram fixed. 



To complete the second diagram, let a function < of 77, be found from 

 the equation 



<f>= 



then it is easily shewn that 



(t<f> 



r tt 



" J 



' 



- 



Hence the first diagram is determined from the second by the same process 

 that the second is determined from the first. They are therefore Reciprocal 

 I>i;igrams both as regards their form and their functions. 



But reciprocal diagrams have a mechanical significance which is capable 

 <!' extensive applications, from the most elementary graphic methods for calcu- 

 lating the stresses of a roof to the most intricate questions about the internal 



