TowNSEND — Geometrical Properties of the Atripht haloid. 67 



JfiV to AB cannot consequently be given in general in such terms. 

 As SJF is never greater than Mli when both are real, and only equal 

 to it when both are evanescent, the least possible value of the ratio of 

 EF io AB given by equation (9), viz., 4 -^ 3, is consequently an in- 

 ferior though by no means a close limit to the possible ratio of 

 MN: AB. But a superior as well as a closer inferior limit, both 

 however much outside the extreme limits of the actual ratio, may be 

 found for it from equation (7) as follows. 



As the volume V of the solid generated by the revolution of either 

 oval round its axis of figure AB must necessarily be less than that of 

 the circumscribed cylinder, and greater than that of the inscribed double 

 cone having AB for axis and the circle described by MN for base, we 

 must therefore have, for all values of a and b, 



1 TT { ^)'(«' - i') < i TT . MN\ AB>-hiT. MN\ AB, 



and therefore, for all values of a and h, as (^a-b) = AB, 



[MNV „ a" +ab + b- a'' + ab + b'^ ,, „, 



from which it follows, consequently, that the square of the ratio in 

 question must always lie between the extreme limits 2 and 8, its least 

 and greatest values corresponding respectively to the least and greatest 

 values 1 and oo of the unrestricted ratio of a to b. From the manner 

 in which they have been obtained, however, these limits are obviously 

 much outside of those of the actual ratio. 



Differentiating equation (3) with respect to x, we get 



y J.= - x + 2hT<?x-^ - 2¥x-^, (11) 



ax ^ 



from which we get, for the values of x- for which y -^ = and for which 



ax 



consequently y"^ is a maximum or a minimum, the cubic equation 



a;8 - 2MV + 2^« = 0, (12) 



which has always one real negative root for which x is consequently 

 imaginary, and whose remaining roots will, by the theory of equa- 

 tions, be real or imaginary according as 87j^- 27^^ is > or < 0, and of 

 the same sign when real. Hence the curve has never more than two 

 values of x"- for which y- is a maximum or a minimum ; and as those 

 values of x^ lie necessarily, one between the limits «;- and V" for which 

 2/" is positive, and the other between the limits ^- and c" for which y~ 

 is negative, there is therefore never more than one pair of maxima 



11. I. A. PllOC, SEE. II., VOL. IV. — SCIENCE. L 



