from a Mechanical Point of View. 699 



squares when the original common difference is made small 

 enough. For example, as a special case of (1), if: 



x + iy = csin(£+^), (2) 



x = c sin f cosh 77, ?/ = c cos f sinh -77 ; 

 and the curves corresponding to 77 — constant are 



x- 



c* cosh 2 7] c* sinh 2 77 

 and those corresponding to f = constant are 



+ «« a ; n v.s M — ^ • • • • w 



* 2 _l!__ 1 / 4 n 



c 2 sin 2 f " c 2 cos 2 £ " ' • ■ • • • W 



a set oE confocal ellipses and hyperbolas. 



It is usual to refer x, y and ^, 77 to separate planes and, 

 so far as I have seen, no transition from the one position to 

 the other is contemplated. But of course there is nothing 

 to forbid the two sets of co-ordinates being taken in the 

 same plane and measured on the same axes. We may then 

 regard the angular points of the network as moving from the 

 one position to the other. 



Some fifteen or twenty years ago I had a model made for 

 me illustrative of these relations. The curves have their 

 material embodiment in wires of hard steel. At the angular 

 points the wires traverse small and rather thick brass disks, 

 bored suitably so as to impose the required perpendicularity, 



Fig. 1. 



the two sets of wires being as nearly as may be in the same 

 plane. But something more is required in order to secure 

 that the rectangular element of the network shall be square. 

 To this end a third set of wires (shown dotted in fig. 1) was 



