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ROYAL SOCIETY OF CANADA 



A convenient way of effecting the transformation is to employ 

 parallel instead of cartesian co-ordinates. The parallel co-ordinates u 

 and y of a straight line are the distances AM, BN, (Fig. 3) of its inter- 



sections by two parallel lines from the origins A and B selected on these 

 parallels. In this system, an equation of the first degree: 



(7) au -\- hv -\- c = o 



defines a point of which the cartesian co-ordinates may be found as 

 follows : Taking 0, centre oî AB, us origin, OB as axis of .t, a parallel 

 through to Ailf and ^iVas axis of y and designating by ô the distance 

 OB, we have : ^ 



b — a 



(8) x= Ô 



(9) 



b-\- a 



c 



y 



b -\- a 



1 Equation (7) gives for u ^ o 



Taking ^(7= ^ and BD^ ^ 



the point defined by equation (7) is P, intersec- 

 tion of A D and B C, (Fig. 4). 



Fig. 4. 



Similar triangles give the following proportions : 



A Q _ Q P 

 AB B D 



and 



B Q _ QP 

 B A AC 



