TRAX8FORMATIOX OF AlHT FIGURE BY 



and r/f is the intercept of the line (17 = const., = const.) cut off between the 

 or&oes and +</ 



The angle t, at which a line whose co-ordinates are functions of p cuts a 

 line whose co-ordinate* are functions of q, is found from the equation 



dx dx dy dy dz dz 



~T~ ~T~ + -J~ -J~ + ~J^ TL 



= C08C 



Expressing this in terms of ij, and , it becomes 



*,*, 



dpdq ' dpdq 



In order that the angle e, at which these lines intersect in the sy- 

 (j-, y, z), should be always equal to the angle e, at which the corresponding 

 lines in the system (f, ij, ) intersect, it is necessary and sufficient that 



a = = y (6), 



for these are the conditions that equations (4) and (5) should be of the sanu- 

 form. 



Now consider the quadrilateral ABCD, cut off from 

 the surface (= const.) by the surfaces f, +d, 77, and 



Since the three sets of surfaces are orthogonal, their intersections are lines 

 of curvature, by Dupin's Theorem. Hence AB is a line of curvature of the 



