( 480 ) 
lt¢,—n’e,_ 2V1+s, 
i; (n—1)? ee ack) 
or 
V+) Ve 
n—l n—l 
So the same condition as had been found above. 
If we represent the condition for the possibility of v >> b, again 
graphically, it is given by a parabola, and that the same as occurs 
in fig. 86 p. 321, but shifted downward in the direction of the 
e,-axis by an amount —1. We need not draw it, but we shall 
think the points of contact with the ¢,-axis and with a line ¢,=—1 
indicated by the letters Q' and P". To satisfy the circumstance 
v>b,, the point (¢,, &,) must lie inside the space which | shall 
call O"P"Q". But for the possibility of the closed figure the point 
(e,, &) must lie inside the space OPQ — in both cases below the 
corresponding parabola. This can only occur when the two areas 
mentioned cover each other or as least overlap. This requires 
(n—1)? >1 or n> 2. So the points (¢,, €,) giving a closed curve, 
for which the value v > 6, occurs between two values of x, are 
confined to a smaller space, again bounded by the axes and a 
parabola. In this case the parabola touches the ¢,-axis at a distance 
n(n -—2) from the origin, but intersects the ¢,-axis at a distance 
n(n—2 n—2 tet ane 
( apes from the origin. The condition that the two values 
n? n 
of x for which v= b,, coincide, and that the closed curve touch a 
line y=), is this: that the point (¢,, ¢,) shall lie on this parabola. 
ie nwe : 
an Land 1 —a«= ——. If we compare this value 
(n— 1) n—l 
of zw with that which we have called z, above, z, appears to be 
Then r= 
dv 
besides highest value of a for which = is equal to 0 for the points 
a“ 
of the closed curve, also the value of z for the point in which the 
closed curve touches the line v=8,. If volumes occur which are 
larger than 5,, then the greatest volume lies at a value of z < ay. 
Let us now more closely examine the space which OPQ and 
0" P'Q" have in common, and inside which the points (¢,, €) must 
lie for the condition v > 6, to be satistied. For very large this 
space will be very large in the direction of the ¢,-axis, but in 
the direction of the ¢,-axis it remains limited to an amount 1—— 
Nn 
and so below unity. Also by simple construction we can now indicate 
