MAXmUJI .VXD MIXIMUM VALUES OF A LINEAR FUNCTION, ETC. 59 



If Q is any point inside the triangle AEF then a>;8 + y and ^o 

 triangle A'B'C exists, but (30) shows obviously that s has a mininumi 

 at A. If § is on A ^, ,8 = 0, a>y and (30) shows s = yh at A the 

 minimum. When Q is at 4^ s = i\ is a minimum there. 



In conclusion, when Q is any point inside the hj'perbolic triangle 

 D E P the function s has a minimum at an ordinary point P inside A B C; 

 when Q is any other point inside A B C s has a singular minimum at a 

 vertex oi A B C ; when Q is D, F ov E s has. pseudo minimum along that 

 side ; when Q is any other point on the boundary of .4 B C s is minimum at 

 the nearest vertex of A B C. 



13. In explanation of the singularities encountered above, let -s be 

 an ordinate to the plane A B C a.t P whose polar coordinates at origin A 

 are p, 0. The nature and character of the singularity of this surface at ^4 

 and the behavior of the function s in that neighborhood are fully explained 

 by the equation to the surface (31). The slope to the plane A B C of the 

 tangent line to the surface at A is 



a -\- j3 cos{p c)-\- y cos(|0 b)= tan ©, 

 and 



s={pc-\-yh)+ ptaji® (35) 



is the equation of a cone of one nappe to the surface at this point. The 

 surface therefore ends in a teat or peak at each vertex of A B C. For all 

 values of a, /S, y that keep this teat pointed down at A there is a mini- 

 mum of s a.t A. The reason of this shape is due to the convention wMch 

 confines r^, r,, r^ to positive values, for if this convention be not observed 

 then s is an 8 valued function symmetrical with respect to the plane ABC. 

 In fact if it be transformed into orthogonal cartesian coordinates and 

 rationalized it will be of degree 8 and of degree 4 in z-. At the points 

 A, B, C this surface has singular points, one of the sheets of wliich has 

 an ordinary conical point at a vertex of the second degree, or a quadratic 

 tangent cone there. Each of the four sheets will in general have a maxi- 

 mum or a minimum ordinate. The separate sheets of this surface will 

 have for ordinates respectively 



ar^ -\- I3r„ -\- yl\. 



ar^ + /3r„ — y;-3, 



ar^ — /3r, -f y?-3, 



— al\ + pr„ + yy,- 



To cliange the sign of two of the r's in the first reproduces one of the 

 others with its sign changed; to change all three changes the sign of the 



