macfaklane: algebra of physics 



367 



circular ratios, differing principally in the line 

 units chosen for the ratio ; and that for the other 

 curves these ratios, differing in conception, may 

 cease to have equal numerical values. For 

 example, PM is said to be drawn perpendicular 

 to OA (fig. 6), or parallel to the tangent at A; 

 and cos AOP is variously given by different 

 authors as OM/OA or OM/OP; while sin AOP 

 with still greater ambiguity is defined as 



- 



F~/?6. 



PM/OA MP/OA PM/OP MP/OP MP/OB PM/OB. 



The simplest ratio is obtained where the two lines of the quo- 

 tient have the same directed unit; thus OM/OA and MP/OB 

 give pure numerical values, whereas OM/OP and MP/OA 

 involve a difference of direction. In the above paper I defined 

 cos AOP as OM/OA, but made the mistake of defining sin AOP 

 as MP/OA. The simple principle mentioned allows definitions 

 to be given of the circular ratios which apply without change to 

 the more complex curves mentioned. 



The principles of elliptic and hyperbolic analysis. Abstract 

 read before the Mathematical Congress at Chicago, August 24, 

 1893. Separately printed. This paper investigates some of the 

 fundamental principles of trig- 

 onometry on the surface of 

 the exsphere, by which is 

 meant the surface of the equi- 

 lateral hyperboloid. Let AQ 

 (fig. 7) be the positive equi- 

 lateral hyperbola, A'Q' the 

 negative, and P"Q" the con- 

 jugate; when the figure re- 

 volves about OA, the whole 

 surface traced out by the three curves forms the exsphere. The 

 angle AOP was defined as the ratio of the area of the sector AOP 

 in square units to the area of the triangle AOB in square units; 



ng.z 



that is u 



a- 



A/~ that is 2 A /a 1 



All the radii from O to the bound- 



