48 'rNIVERSITY OF VIRGINIA PUBLICATIONS 



Let A B C be a given triangle and P a point in its plane at distances 

 )\, )■.,, r., irom A, B, C. We seek the maximum and minimum values of 

 the function 



s = Li\ + Mr., + Ni\, (1) 



where L, M . X are arbitrary' real constants, and i\, r„, r^ are absolute 

 numbers called the radial coordinates of the point P vnth respect to the 

 fixed triangle .4. B C. 



The problem is simplified by considering L, M, N to either be all 

 positive or only one of them negative. For if they be all negative it 

 amounts to considering — s when they are all positive. If any two of 

 them, say L, M be negative the case is in like manner reduced to the 

 investigation of 



— {Li\ + Mr„ — Nr^) . 



Moreover we may always take L, M, N so that their sum shall be 

 unity, for if not so then on di^dding both sides of (1) by the sum of 

 L, M, N the case is reduced to this condition, except when the sum of 

 L, M, N may be zero, a condition to be considered subsequently. We 

 therefore consider 



s = ar, + /3r, + yr„ (2) 



wherein a + j8 + y ^ 1. and only one of them may be negative. We may 

 consider a, p. -y to be the area! coordinates of a point Q with reference 

 to the triangle A B C. 



4. The necessary conditions.^ — With reference to an arbitrarily chosen 

 system of orthogonal cartesian coordinates, let P be x^ y and A, B, be 

 respectively .r,, y^; a\, y^', x„, y„. The sign of the radical being taken 

 positive 



>\ = V(.T — a:,)=+ {y — y^y-, (3) 



where the eqiiation of P .4 is 



x — x^ y — y, 



— -. = = i\, (4) 



?,, /»! being the direction cosines of P A. We have like equations for 

 r„, ('3 on changing the subscripts. 



Eqiiating to zero the first partial derivatives of 5, in (2), with respect 

 to X and to y, there result the necessarj^ conditions for an ordinary 

 maximum or minimum of s 



