EXAMPLES OF MAXIMA AND MINIMA, 257 



sent any four consecutive sides. Then, by what we have 

 just seen, P, Q, R, 8 must lie on the circumference of a 

 circle : for otherwise the area could be increased, by leaving 

 the rest of the figure unchanged, and shifting PQ, QR, RS 

 until the points P, Q, R, 8 did lie on the circumference of a 

 circle. Similarly Q, R, S, T must lie on the circumference of 

 a circle. And this circle is the same as the former circle, for 

 it is the circle described round the triangle QRS. In this man- 

 ner we shew that when the area is greatest the figure must 

 have all its angular points on the circumference of a circle. 



Suppose an indefinitely large number of consecutive sides 

 of the figure to become indefinitely small: then the cor- 

 responding portion of the boundary of the greatest area be- 

 comes an arc of the circle of which the remaining sides are 

 chords. Hence we obtain the following general result : if an 

 area is to be bounded by given straight rods and strings, the 

 area is greatest when the strings are all arcs of the same 

 circle, and the straight rods all chords of that circle. 



The following problem is analogous to that which we have 

 been considering. Required to determine the greatest area 

 which can be inclosed by a quadrilateral three of whose sides 

 are given. 



Let a, 6, c denote the lengths of the three given sides, 

 taken in order of contiguity. Let 6 denote the angle between 

 the sides b and c, and </> the angle between the side a and 

 that diagonal which passes through the angle between a and 

 6. Then the area of the figure is 



- be sin 6 -f - a \/(b z + c* 2bc cos 6} sin <. 



2 2 



This is a function of the two independent variables 6 and <f> ; 

 but we can obtain the result which we require without going 

 through the usual process for finding the maximum value of 

 a function of two independent variables. For we see that 

 to ensure the greatest area < must be a right angle. In a 

 similar manner we might shew that the angle between the 

 side c and that diagonal which passes through the angle 

 between 6 and c must also be a right angle. Hence the qua- 

 drilateral figure must be capable of being inscribed in a circle 

 of which the side not given must be the diameter. 



T. D. C. S 



