PROJECTION IN TWO DIMENSIONS. 15 



extremities of the minor axis. By the aid of this curve 

 we may readily obtain any ellipse which may be derived by 

 projection from a given circle — any line through the centre 

 and terminated by the external branches will be a major 

 axis ; to obtain the corresponding minor axis, draw a line 

 at right angles, then the portion intercepted between the 

 internal branches will give the magnitude of the minor 

 axis. 



The equation to the curve in rectangular coordinates is 



its area is two thirds the area of the primitive circle. 



As previously stated, we have some choice of method in 

 constructing a projected curved area ; in (6) the elemen- 

 tary rectangles have been so piled up that their centres lie 



on the line 



_ m ma 



that is, on a line parallel to the primitive axis. If the 

 locus of the middle points Avere the line 



m ma 



^/= -,x-\ J, 



•^ 2/ 2/ 



we should obtain an equation of the form 



m^[x — aY + \ml{x — a)y-\- (4— 3m^)?/^ = wiV, 



representing an ellipse of which the perimeter is equal 

 to the perimeter of the primitive circle. If any line 

 y^h cut this ellipse, the length of the section will be 

 2^m^c^ — h^; this will also be the length of the section 

 made by the same line with (6) . 



Inverse Problems in Projection. — In the foregoing 

 remarks it has been supposed that the primitive curve 

 has been given and the projectrix obtained by means of 



