DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 315 
Instead of the usual symbol, 1 — sin^®=\/ 1 — ? sin^ip, \/I has been substi- 
tuted, when i is the modulus. Should it become necessary to designate the ampli- 
tude, the expression may be written hp, or ^1. 
For the elliptic integrals of the first and second orders, which are usually written 
TXP) And Ee(?>), I have substituted And fd(p\/l. The surface of revolution 
may be named the generating surface, while the intersecting surface is always a cylin- 
drical surface. The parameter, of which p is the general symbol, we shall suppose 
to vary from positive to negative infinity, and to pass through all intermediate states 
of magnitude. 
The nature of the representative curve will depend on the value assigned to the 
parameter p in the expression K (V- . „ -o • « • The modulus we shall assume 
J [ +J9 V 1 — sin'^ip 
to be invariable and less than 1 . In this progress from -boo to — oo, the parameter 
passes through thirteen distinct values, each of which will cause a variation in the 
species or properties of the hyperconic section, the representative curve of the given 
elliptic integral. 
In the following Table we may observe that the generating surface in passing from 
a sphere to a paraboloid, in its state of transition, becomes a plane. 
It is somewhat remarkable, that the common form of the elliptic integral of the 
first order does not appear in the Table, although it is implicitly contained in cases 
II. and VIII.; for in the circular form of the third order, when the parameter is 
equal to the modulus i, we can reduce the third order to the first. The reason why 
the first form of elliptic integral does not appear in the Table is this ; in the thirteen 
cases given, the origin is placed at the centre, or symmetrically with respect to the 
represented curve. When the elliptic integral of the first order is given in the usual 
form, without a parameter, it represents a spherical parabola, but the origin is non- 
syrnmetrical, that is, the origin is placed at a focus. See Theory of Elliptical In- 
tegrals, p. 3.3. 
Instead of p, the general symbol for the parameter, we may substitute for it parti- 
cular values, such as /, yn, or n, as the case may require. The quantities /, m, n, i and 
j, are connected by the following equations ; — 
P-\-f=\, hn—i^, and m — 72-l-mw=P, in the circular form, “j 
ln=P, and m-t-n — 7n/z=:?, in the logarithmic form,/ 
m and n may be called conjugate parameters ; while I and m, or I and n may be termed 
reciprocal parameters. 
These thirteen cases are exhibited in the following Table. 
