186 

 Making these substitutions, 



and if X, /u, v are the angles which a semidiameter r through one of 

 the points of contact makes with the axes, 



r r r r 



Similarly, for another point of contact and semidiameter r', we have 



cos X'= ^, cos p'= &, cos v'= ^ ; 

 r r' r 1 



whence 



cos X cos X' + cos fj. cos // + cos v cos v 1 = 0, 



or r and r t are at right angles. 



But facility of proof is not the sole advantage of this method. It 

 enables us to bring prominently into view that great principle of 

 duality which is involved in all our geometrical investigations. This 

 principle may be familiarly stated in the form, that every geometrical 

 theorem or mathematical truth has its double. As an illustration of 

 this, let us take the tangential equation of the surface of curvature, 



- 1), 



and instead of , v, , write down x t y, z t introducing the constant r 

 to render the equation homogeneous, and it becomes 



-,-<). . (14) 



Now this surface has properties which are one by one reciprocal to 

 those of the " surface of centres." As, for example, in each of the 

 principal planes, the sections of the surface are ellipses and curves 

 whose equations are of the form A 2 o? 2 -fBy=a? 2 y 2 . 



In the mean section the ellipse will touch this curve in four points, 

 through which four lines being drawn parallel to the axis of y, they 

 will lie wholly on the surface. 



The formulae which exhibit the relations between the protective 

 and tangential coordinates of the same curve or curved surface are 

 simple and symmetrical. They are given here without demonstration. 



Let <I>=0(, v, ) = he the tangential equation of a curved sur- 



