72 REPORT—1846. 
<s, The problem of the division of its perimeter is accordingly a geome- 
trical interpretation of that of the division of the complete integral, and was 
considered by mathematicians at a time when the theory of elliptic func- 
tions was almost wholly undeveloped. Besides Fagnani, whose researches 
with respect to the lemniscate have been already noticed, we may mention 
those of Euler, who however did not succeed in obtaining a solution of the 
problem. Legendre, who seems to have attached considerable importance 
to geometrical illustrations of his analytical results, assigned the equation of 
a curve of the sixth order, whose arcs measured from a fixed point represent 
the sum of any elliptic integral of the first kind and an algebraical expression. 
He showed also that an arc of the curve might be assigned equal to the el- 
liptic integral, but in order to this both extremities of the are must be con- 
sidered variable, so that in effect the integral is represented by the difference 
of two ares measured from a fixed point (Traité des Fonctions Elliptiques, 
i. p. 36). 
IM. coe in a note presented to the Institute in 1843 (Liouville’s Journal, 
viii. 145), has proved a beautiful theorem, viz. that the sum and difference of 
the two unequal arcs, intercepted by lines drawn from the centre of Cassini’s 
ellipse to cut the curve, are each equal to an elliptic integral of the first 
kind, and that the moduli of the two integrals are complementary. In the 
lemniscate, which is a case of Cassini’s ellipse, one of these arcs disappears, 
and the moduli of the two integrals are equal, each being the sine of half a 
right angle. So that M. Serret’s theorem is an extension of the known pro- 
perty of the lemniscate. 
M. Serret has since considered the subject of the representation of elliptic 
and hyper-elliptic ares in a very general manner. His memoir, which was 
presented to the Institute and ordered to be published in the ‘ Savans 
Etrangers,’ appears in Liouville’s Journal, x. 257. He had remarked that 
the rectangular coordinates of the lemniscate are rationally expressible in 
terms of the argument of the elliptic integral which represents the are, 
_ 3 is — z3 
for if we assume «= VW Baza and y= V7 2a, we shall have 
, and if between the first two of these 
dz 
— 2 2 =—2 es 
ds=V {dx +dy}= oe 
equations we eliminate z, we arrive at the known equation of the lemniscate*. 
So that if we state the indeterminate equation 
da +dy=Z.dz2*, 
(2, y and Z being real and rational functions of z), the lemniscate will afford 
us one solution of it; and every other solution will correspond to some curve 
whose arc is expressible by an elliptic or hyper-elliptic integral. Of this in- 
determinate equation M. Serret discusses a particular case. He succeeds in 
solving it by a most ingenious method, which is applicable to the general 
equation, and shows from hence that there are an infinity of curves, the ares 
of which represent elliptic integrals of the first kind. M. Serret’s researches 
however have not led him to a geometrical representation by means of an 
algebraical curve of any integral of the first kind, though his results are ge- 
neralised in a note appended to his memoir by M. Liouville. In order that 
* On reducing the integral we ae to the standard form of elliptic integrals, we 
find that it is an elliptic integral of the first kind, of which the modulus is the sine of 45°. 
