46 Trans. Acad. Sci. of St. Louis 



The area of this ellipse varies therefore as cos a. We have 

 then special cases as follows : — 



a = 0° A — ttR 2 Circle Maximum area. 



a = 90° A = Straight Line Minimum area. 



It will be seen from the above equation for the area and 

 from the transformed areas of other figures that the area of 

 the transformed figure is obtained by multiplying the original 

 area by cos a. 



It will also be observed that the area of the transformed 

 figure is the same as that of the orthographic projection of 

 the original figure on a plane at angle a with the plane of the 

 original figure. 



Isogonic Transformation may be applied equally well to 

 solids. 



Let us take its application to a sphere (Fig. 3) referred to 

 the rectangular axes OX', OY, OZ', whose equation is 



x' 2 + y' 2 + z 2 = R 2 . (5) 



Let P' be any point on the sphere and let the co-ordinates 

 of P' be x', y' , z' . 



Turn the ordinate z' through an angle a about its foot, 

 keeping it always parallel with the X'Z' plane. P' will 

 go to P whose co-ordinates are x t , y x , z v 



Now, turning the co-ordinate system backward through an 

 angle 6 (to be determined later) about OY' we have xyz as 

 the new co-ordinates of P referred to OX, OY, OZ. 



The equations for the first transformation are 



x' = x l + z x tan a 



V' = Vi 



z' — z 1 sec a 



and equation (5) becomes 



(iCj + »! tan a) 2 + y? + (z t sec a) 2 = R 2 

 or 



x 2 + 2x x z l tan a + y 2 + z 2 + 2z 2 tan 2 a = R 2 (6) 



