MAXl.Mril AXD illXIJtril VALncS OF A LINICAn FUXCTIOX. ICTC. 



53 



is equal to the power of the point .r, y, z with respect to the self-polar 

 circle oi A B C. It will be observed that s = when Q is at either of the 

 excenters of A B C, for then 8 = A and the power of each of these points 

 with respect to the self-polar circle is 8 A^. Hence the circle passing 

 through the excenters of ABC is concentric with the self -polar circle 

 ha\ing the ortho-center oi A B G for center. At z.txj point Q on the 

 circumference of this circle, which satisfies the necessary conditions for 

 ordinary maximum, tlie value of s is zero; it will develop sul)sec|uentl)-. 

 however, that only the excenters on this circle satisfy the ordinary con- 

 ditions. 



The values of the radial coordinates of P can now be determined from 



-a-)h ± {c'- -\- h'-~a'-)\ 



28s 



(25) 



2 A sin(.-l' ±.-1] 

 s sin .4' sin A 



interchange of letters giving 1\ and i\. 



The equations thus determined furnish the radial and the areal coordi- 

 nates of the critical point P and the value of s there for any given real 

 numbers L, M, N in terms- of a, p, y. We now examine the sufficient 

 conditions with the view of determining whether .s is a maximum or a 

 minimum. 



8. The sufficient condition. — The second partial derivatives of 5, in 

 (2), with respect to x, y, a- and y, are found to be respectively 



dxdy 



(26) 



