72 Journal of the Mitchell Society [December 



and give a rectangle of positive area, they derive from the rejected 

 value ? = 135°. The two points 

 + a _ — b 



and 



— a + b 



-^ = 7t' ^ = VT 



neither formally satisfy the fundamental function 



ab 



nor derive together from either 9 = 45° on ? = 135°. 



The association of the ellipse and the rectangular hyperbola in 



this problem suggests the query whether the maximum rectangle 



inscribed in the elUpse is at the same time the maximum rectangle 



which may be inscribed in the rectangular hyperbola. This question 



must obviously be answered in the negative, since the rectangular 



ab 

 hyperbola xy = — - at once furnishes the result 



U = 4xy = 2ab; 



which shows the in variance of the rectangle area for the rectangular 

 hyperbola, as indicated by the nomenclature. 



The superiority of Method V over those which precede is con- 

 spicuous in two respects — the brevity and elegance of the treatment, 

 and the immediate identification of the critical values which give 

 rise to a maximum value. 

 Chapel Hill, N. C. 



