66 Proceedings of the Royal Irish Academy. 



'E3. — The funetion in question being, by the theory of equations, 

 the product of the squares of the differences of the roots of equation (1) 

 for the case when co = 0, the same value for it in terms of a and I would 

 be obtained even more readily than above by the substitution for c in 

 that product of its value in terms of a and I. See equations (1) 

 and (5). 



Multiplying both sides of equation (3) by tt^^, integrating between 

 the limits a and I, and substituting in the result for h and h their 

 values just given in terms of a and l ; we get, for the volume Fof the 

 solid generated by the revolution of either oval round its axis of figure 

 AB, the value in terms of a and I, viz. — 



which, compared with that of the volume S of the sphere on AB as 

 diameter, viz., \ tt [a - hf, gives for the ratio of the two volumes in 

 terms of a and h the value 



V , «■-+ ah -I- ¥ .„. 



a value which, lying always between the extreme limits 3 and 4 cor- 

 responding respectively to the extreme values 1 and oo of the unre- 

 stricted ratio of a to h, shows, consequently, that the extreme depth 

 MN is always greater than the extreme breadth AB of the ovals 

 (see fig.) 



That the chords of contact 1!F of the tangents to the ovals from 

 the centre of the curve (see fig.), which when the ovals are finite 

 are of course always less than their extreme depths MN, are also 

 in all cases greater than their extreme breadths, AB, may be readily 

 shown from equation (6) as follows. Since 



therefore from equation (6), as («; - 5) = AB, and as Jil^ = a-b-{a- + ab + b'-) 

 ■f («+ b)~ by equations (5), 



fEFy 4 [2 {a- + ai + ¥) + 3ahJ 



[aBJ ~ 9 {a- + «5 -1- ¥) {a + hf ' ^^^ 



a ratio always exceeding unity, and having the extreme values 27 -=- 9 

 and 16 -f 9 for the extreme values 1 and co of the unrestricted ratio of 

 a to b. 



The value of MW in terms of a and b not being in general deter- 

 minable like that of FF in finite terms, the exact value of the ratio of 



