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Prof. CayJey on the Plane Representation 



the distance from the inverse square to the simple inverse. I 

 forbear going further into this subject at present, and shall only 

 remark further that these views bear in an important manner 

 on the explanation by theory of certain phenomena of light, and 

 on other parts of theoretical physics. 

 Cambridge, March 14, 1871. 



XXXVIII. On the Plane Representation of a Solid Figure. 

 By Professor Cayley, F.R.S* 



WE represent in piano the position of a point P whose co- 

 ordinates in space are {x, y, z), by drawing these coordi- 

 nates, on the same scale or on different scales, and in given 

 directions from a fixed origin in the plane; OM=a?, MP' = y, 

 P f P"=^. But observe that the point P" alone does not com- 

 pletely represent the point P ; in fact P" represents a whole 

 series of points lying in a line ; any one such point is the point 



'M 



whose coordinates are m, mp 1 , p 1 P /f . For the complete repre- 

 sentation of P we require the two points P', P" : these might be 

 distinguished as the projection P", and the foot-point P'. The 

 two points P', P" are obviously such that the line joining them 

 is in a given direction. 



The preceding is, of course, the ordinary method of orthogonal 

 projection, or geometrical delineation of a solid figure : it may 

 be used under various forms ; for example, the coordinates x, y, z 

 may be taken on the same scale and in directions inclined to 

 each other at angles of 120° (isometrical projection) ; or the 

 coordinates oc, y jnay be drawn on the same scale and at their 

 actual inclination, 90°, to each other ; and the coordinate z on the 

 same or an altered scale in any given direction ; the points P' 

 then give a true ground-plan of the solid figure, and the lengths 



* Communicated by the Author. 



