256 EXAMPLES OF MAXIMA AND MINIMA. 



From (1) and (2), 



-j- = a/3 cos xDx + 78 cos yDy (3), 



= a/3 sin xDx 78 sin yDy (4), 



f dd> n ( sin x cos y) ,, 



therefore - = a/3^cosaH = -\ Dx (5). 



at { smy ) 



Hence, since the coefficient of Dx must vanish, 

 sin (x + y) = 0. 



Therefore x + y must be zero, or some multiple of TT; the 

 only solution applicable to the present question is 



r i 7 , _ _ />\ 



^ ^ y v \vj. 



Hence cos y = cos x : substituting this value of cos y in 

 equation (2), we have 



cosar = 



a- -u /r\ a sin r 



. Since by (5) -- = ^ - Dx, 



-- 

 dt 



we have, neglecting such terms as vanish, by (G), 



c? 2 <f> aj8 cos (x + y] 

 ~-nt=- - - 

 dt* sin y 



which, by means of (4) and (6), becomes 



sin y \ 

 Hence, since -~ is negative, we have found a maximum 



Cat 



value of <, namely, when the sum of two opposite angles of 

 the figure is equal to two right angles. 



Thus the quadrilateral must be capable of being inscribed 

 in a circle. 



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

 lineal figure are given the area is greatest when the figure 

 can be inscribed in a circle. For let PQ, QR, RS, STrepre- 



