DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 69 
XCIV. The general fundamental expressions for the rectification of curve lines, 
whether of single or double flexion, show that the arc of a curve may in general be 
represented as the sum of two quantities, an integrated and a non-integrated part, 
or as the proposition may be more briefly put, an arc of a curve may be expressed 
as the sum of an integral and a residual. Thus the arc of a plane ellipse is equal 
to an integral and a residual, which latter is a right line. An arc of a parabola is 
the sum of an integral and a residual, which latter is also a right line. An arc of 
a spherical ellipse is the sum of an integral and a residual, the latter being an arc of 
a circle, while an arc of a logarithmic ellipse is made up of two portions, one a sum 
of integrals, the other — the residual — being an arc of a common parabola. It 
appears therefore to be an expenditure of skill in a wrong direction to devise curves 
whose arcs should differ from the corresponding arcs of hyperconic sections by the 
above-named residuals. Thus geometers have sought to discover plane curves whose 
arcs should be represented by elliptic integrals of the first order, without any residual 
quantity — the common lemniscate for example, when the modulus has a particular 
value. It is possible that such may be found. In the same way, an exponential* curve 
C d6 
may be devised, whose arc shall be represented by the integral instead of 
taking it with the residual quantity k tanfl secfl, as the expression for an arc of a 
common parabola. Thus geometers have been led to look for the types of elliptic 
integrals among the higher orders of plane curves, overlooking the analogy which 
points to the intersection of surfaces of the second order as the natural geometrical 
types of those integrals. 
* The equation of this exponential curve is e* cos^J = l. It is easily seen that when ,r=0, y — 0, also. 
kit 
And when x——, y=co . Hence the curve passes through the origin and has asymptots parallel to the axis 
• kit 
of y at the distance — from the origin. 
2 
If we substitute for cos its exponential expression 
y+xd- 1 ~ I |~ y— ,tV-i 
I e" k — 
e^-i + e 
; the equa- 
tion of the curve becomes 
+ 
= 0 . 
e k — 1 
J L 
The common equation of the circle y' + x'—kr, may be written 
log 
[ 
y + x^Z — 1 
= 0 . 
Fig. 29. 
In this form the similarity of the equations of the exponential curve and the circle is 
evident. 
y , . CdO . 
In the equation if we make the imaginary transformation tan0= v _i sinco, the resulting ex- 
pression will be s=kto ^ or the expression is transformed from a logarithmic to a circular function. 
