( 482 ) 
above as follows. When we have a point (¢,, ¢,) in the space which 
OPQ and O"P'Q" have in common, the closed curve will possess 
volumes which are greater than 4, — and by shifting this point 
in the direction of the axis of the parabola till it meets the first- 
mentioned shifted parabola, we find the value Be) (n—1)’, in the 
projection of this displacement on the ¢,-axis, or the value of 
(@,—2,)? (n—1)? 
4 n° 
So the length of the line drawn through the given point in the 
direction of the axis of the parabola till it meets the second parabola 
teaches us the value of (z,—a,)* ; to which we may add that the same 
line prolonged to the other side so below the given point, shows 
us also at what value of a the middle of x, and x, lies. If the 
continuation of this line passes through the point ¢, — O ande, = — 1, 
in the projection of this displacement on the ¢,-axis. 
1 ete 
the middle of 2, and 2, lies exactly at Gc If this line intersects 
; ete 1 
the ¢«,-axis below e, = — 1, then a ee and the other way 
about. We have viz. from (9): 
il = pst 
#,+2@#,=14 ple ne 
(n—1)? 
bP re ae ae 
or putting aan Lm 8 
1+¢,—n’e, 
1 — 22 — ree we . 
For given value of x, this represents a straight line, the direction 
é : : : : : 
of which is given by — =n’. This straight line intersects the €,-axis 
é, 
in a point «, + 1 = — (n—1)’ (1—2z,,); from this formula the given 
rule appears. 
Such rules may also be given for the dimension and the place of 
the closed curve itself — and for the accurate knowledge of the 
properties of this curve the knowledge of such rules is not devoid 
of importance. Thus the equation (3) of p. 319 Contribution X 
leads to: 
(z,—«,)* == 
when the values of z between which the curve exists, are represented 
by a, and z,. If we derive from this : 
