54 UNIVERSITY OF VIRGINIA PUBLICATIONS 



On substitution and proper reduction the discriminant 

 d'-s d-s / 8-s 



dx- By- \ dx By 



can be written 







2 



"^ sin=(?v,). (27) 



Using (11) and (13) the discriminant may be written 

 48-s* xyz 



otherwise as 



48= s 



(28) 



Wlien the discriminant is negative s is a maximum, when positive a 

 minimum. 



9. We now examine more closely into the positions of P with respect 

 to Q, considering first the case when Q is inside ABC. Here a, /?, y are 

 all positive and Q is confined to the triangle D £^ i^^ or on a side or at a 

 vertex of this triangle. Since x, y, z must also be positive or rather cannot 

 be negative P is confined to the interior ot A B C qy its boundary. The 

 value of s is always positive when a, p, y are positive and (29) shows the 

 discriminant to be positive and s a minimum. 



Examination of (21) and (25) show that if the position of Q is such 

 that the sum of the corresponding angles of A'B'C and A B G is less than 

 TT then X, y, z are definite positive numbers, as are also rj, r„, r^, and s has a 

 minimum value at this point P inside A B C. 



10. If, however, a pair of these corresponding angles be supplemental, 

 for example 



A' + A= TT, 



then r^=0, r^^^c, i\^='b, also x=l. y=0, z=0. Hence P is at A. The 

 direction cosines of Ci in (4) and (5) are indeterminate. The partial deriv- 

 atives in (5) and (26) are indeterminate. The values of these derivatives 

 depend, therefore, on the path of approach of P to A and the direction in 



