268 



MOREY 



ART G 



dp _ 

 dt 



7,' 1 X' 



v" 1 



y 



x" y" 



■n'" 1 x'" y'" 



ly ^ jy r,jy 



T) i. X y 



v' 1 

 v" 1 

 '"I 



y 



V 



V 



ly 



X 



x" y" 

 x'" y'" 



1 x^'^y'"' 



in which composition is represented by the weight fractions 

 a:, ?/, and \ — x — y oi the three components. Expansion of the 

 right-hand side of this equation gives 



(15) 



The coefficients of -q', 77", v' , v", etc., represent the areas of the 

 triangles p"p"'p^^, prpr„piv^ P'P"P^^, and P'P"P"', re- 

 spectively. It is important to bear in mind the direction in 

 which a given triangle is circumscribed, since, if the area of the 

 triangle P'P"P"' is positive, that of the triangle P"P'P"' is 

 negative. 



Since the above coefficients represent areas, we will denote 

 the determinants by the letter A, followed by subscripts indicat- 

 ing which triangle is meant, and the direction in which it is 

 circumscribed is given by the order of the subscripts. Thus 

 A 123 represents the determinant 



1 x' y' 

 1 x" y 

 1 x"'y'' 



the area of the triangle P'P"P"' 



II 



The equation becomes 



dy 

 dt 



AiSiV ~ A 134 17" + Ai2iV"' — A 123 V 



A23iV' — Amv" -\- AmV 



AmV 



IV 



