74 Mr. H. Hilton on tlie (Jraphical 



circles <] and q' cuts them in H, K ; H^, K', then any circle 

 cutting g, q' at equal angles is orthogonal to the circle ^i 

 coaxial with q and q\ ^vith respect to which H and K' are 

 inverse points ; and any circle cutting q^q' at supplementary 

 angles is orthogonal to the circle to coaxial with q and </', 

 with respect to which H and H' are inverse points. This is 

 readily proved by inverting q and q' into concentric circles. 

 In the present case (in which the two lines on the cloth are 

 arcs of q and q') the circle t^ is imaginary, but SiYi is the 

 real radical axis of" fj and ACBD. 



These eight examples will serve to show that we have in 

 the stereographic net a useful help for obtaining approximate 

 solutions of certain types of problems, and for checking the 

 results obtained by more tedious, if more accurate, methods. 

 It is also a device of the highest educational value. Moreover, 

 these o-eometrical methods often suoro-est the solution of 

 algebraical or trigonometrical problems : for instance, the 

 discussion on pp. 72-73 shows that the values of 7 for which 



, /sin rt + sin X, . cos 7\ _ , , sin 3 -J- sin X . cos y\ 



COS-l r -. ' ) + COS-1 ( -. '- ) 



V COS X . sm 7 / \ COS \ . sm 7 / 



have their miuimum values are 



cos-M — smX .1 sm^-^ sec — -^j j- . 



Since many astronomical problems depend on the solution 

 of spherical triangles, we shall conclude with a description of 

 the method of using the net for the solution of such triangles*. 



(1) Given the three sides. 



The procedure is the same as that employed in solving the 

 7th astronomical problem. 



(2) Given the three angles. 



The easiest method is to find the angles of the "polar 

 triangle"" whose sides are given, and thus deduce the sides of 

 the original triangle. 



(oj Given two sides and the included angle. 



Trace throuorh P. coincidino- with A. the arc ADB and 

 that one of the lines m which makes an angle with ADB 

 equal to the given angle. Take PQ^ PR along the two 

 traced lines equal to the two given sides (fig. 2). Join QR 

 by a circle which is the projection of a great circle, and 

 measure the length QR and the angles PQR and PRQ (see 

 p. 68). 



* See also Penlield, loc. cif. p. llo. 



