258 EXAMPLES OF MAXIMA AND MINIMA. 



It may now be shewn that when all the sides of a recti- 

 lineal figure are given except one, the area is greatest when 

 the figure can be inscribed in a circle of which the side not 

 given is the diameter. 



For let QR represent the side not given, and PQ an adja- 

 cent side. Then the whole figure must be capable of being 

 inscribed in a circle : for otherwise the area could be increased 

 without changing the length of any side. And the angle 

 QPR must be a right angle : for otherwise we might leave 

 PQ and PR unchanged, and by changing QR replace the 

 triangle PQR by a larger triangle. And since QPR is a right 

 angle, QR is a diameter of the circle surrounding the figure. 



3. Find the maximum and minimum value of u* when 

 tt ' = aV + &y + cV ..................... (1), 



while x a + y* + z* = 1 ........................ (2), 



and Ix + my + nz = ........................ (3). 



From (1), (2), and (3), we deduce 



Q = a*xDx + tfyDy + c'zDz .................... (4), 



0= xDx + yDy + zDz ...................... (5), 



0= IDx + mDy +nDz ...................... (6). 



Multiply (5) by X, and (6) by \ and add to (4) j then 

 equate to zero the coefficients of Dx, Dy, Dz thus 



= ..................... (7), 



*=o ..................... (8), 



c'z +\z + \n = ................ ..... (9). 



Multiply (7) by x, (8) by y, and (9) by z, and add : then 

 by (2) and (3), 



Hence 



