58 UNIVEESITT OF VIRGINIA PUBLICATIONS 



Therefore, for any position of Q inside tlie hyperbolic triangle D E F 

 the function s has a determinate ordinary minimum at a point P inside 

 the triangle ABC. When Q is on the boundary of this triangle s has a 

 singular minimum at the corresponding vertex of A B 0, and when Q 

 is at a vertex ot D E F s has a pseudo-minimum on the corresponding side 

 of ABC. 



11. Suppose now that Q is some point in the triangle D E F but 

 between the hyi3erbolic arc EF and the chord E F: Then A -{- A'^-n-. 

 The analytical conditions (5), (11), (13) fail to give a point P of ordinary 

 position at which s has a maximum or minimum. For p^ is negative, p, 

 and Ps positive as shown by (19). Under the condition that the r's are 

 positive s is positive, while (35) shows r^ negative and r^, r.^ positive, also 

 (21) shows X, y, z unlike signed. Hence there is no ordinary point in 

 the plane which fulfils the condition that a, ^, y applied there and directed 

 toward A, B, G shall be in equilibrium. There must, however, be a mini- 

 mum value of s at a finite point in the plane since s is a one valued positive 

 continuous function which is infinite when the rs are infinite. The method 

 of the latter part of § 10 applies here, for a. ^, y are the sides of a 

 triangle A'B'C. Let A -{- A' equal tt -\- 9, then 



-1' = TT — {A — e) <-K. 

 and ^<-4. Hence (34) becomes 



a^ = /32 -f y2 _^ 2Py cos(A — 6). 

 Subtract (33), and 



a2_t,2 = 4;8y sin(4— |^)sin^e, 



is always positive and a>v. Hence s is a minimum at the singular point 

 A, as in the previous case. This maj'' also be briefly shown as follows : 

 Let Q be in the position assigned above. The straight line 



xi\ + yr^ + zr^ = ar-, + /Jr^ -f yv^, 

 for any assigned position of P, passes through Q and cuts the hyperbolic 

 arc E F ina. point x', y', z' . It has been shown in § 10 that 



2"' ^'i + y' i\ -\- ^' i\ '>P c + yi, 

 hence s = ar.^-\- pr.^ -\- yr^ is a minimum at A. 



Now let Q be any point on the chord E F. Then a = ,8 + y. The 

 point at wliich these segments directed to A, B, C are in equilibrium is at 

 infinity. However, the coeflBcient of p in (30) is at once obviously posi- 

 tive and s has a minimum at A. 



