[From the Proceedings of the London Mathematical Society, Vol. IV.] 



L. On the Condition that, in the Transformation of any Figure by Curvilinear 

 Co-ordinates in Three Dimensions, every Angle in the new Figure shall be 

 equal to the corresponding Angle in the original Figure. 



[Read May 9th, 1872.] 



IN the corresponding problem in two dimensions, the only condition is 



x+J ly =/(+</ 177) (1), 



where x, y are the co-ordinates in one system, and 77 in the other. 

 In three dimensions the solution is more restricted. 



Let x, y, z be functions of 77, ; then the point in the system x, y, z, 

 for which is constant, will be in a certain surface, and by giving a series 

 of values we obtain a series of such surfaces. There will be two other series of 

 surfaces, corresponding to 77 and respectively. If in the second system , 77, 

 are rectangular co-ordinates, the surfaces corresponding to 77, in the first 

 system will intersect at right angles. The condition of this is 



dx dx dy dy dz dz _ 



~j 7 b T~ ~T TZ "y" ~~i TTv ~~ *J 



(2), 



with two other equations in and and and 77. 

 If we now write 



o = 



'dx 



'dx 



\' + t*y\' + (*\' 



/ \dr)/ \drj) 



HD'+f 



.(3); 



VOL. n. 



38 



