210 CASES OF GEOMETRICAL MAXIMA AND MINIMA. 



axes, whose sum is a minimum. By a different method, we 

 might have obtained at first the minimum value of r + r'. 



For since r 8 + r' 8 = a 2 + 6 2 , and rr sin = ab, 

 wehave (r + r'Y = a'+ 



Sill v 



where is the angle between r and r'. Differentiate with 



2ab cos ~ , , ,, ,, TT , . 



respect to 6, and we get -- ,~2~/r~ = "> therefore = - ; this 



sin (/ .. 



7/1 



gives the minimum value as before ; -5-7 = would give us a 

 second result, which would be the maximum. 



The foregoing Article has been derived from the third 

 volume of the Cambridge Mathematical Journal, page 237. 

 The following problem will furnish an exercise. Find the 

 maximum or minimum length of the straight line drawn from 

 the end of the minor axis of an ellipse to meet the curve. 

 If x, y, be co-ordinates of the point where a straight line 

 drawn from the end of the minor axis meets the curve, the 

 length of the straight line can be expressed either as a func- 

 tion of x or of y ; thus two solutions can be obtained and 

 compared. 



In the solution of some of the examples on maxima and 

 minima the following results will be required : they may be 

 established by means of the Integral Calculus. 



The volume of a right cylinder is found by multiplying 

 the area of its base by its altitude. 



The convex surface of a right cylinder is found by multi- 

 plying the perimeter of its base by its altitude. 



The volume of a right cone is one-third of the product of 

 its base and altitude. 



The convex surface of a right cone on a circular base is 

 one-half the product of its slant side and the perimeter of 

 its base. 



If r be the radius of a sphere its volume is - - and its 



o 

 surface is 47TT 2 . 



