r 
i ON THE RECENT PROGRESS OF ANALYSIS. 45 
egendre’s own methods, and he deduces from it the remarkable conclusion, 
hut if of a series of ellipses, whose eccentricities are connected by a certain 
law, we could rectify any two, we could deduce from hence the rectification 
of all the rest. The law connecting the eccentricities of the ellipses is that 
which would be obtained by making use of Lagrange’s method of transfor- 
mation, with which accordingly this result is closely allied. 
Legendre’s next work was an essay on transcendents *, presented to the 
Academy in 1792 and published separately the year after. It contains the 
same general view as that which is developed in the first volume of the 
‘ Exercices de Calcul Intégral,’ which appeared in 1811. 
12. The theory of elliptic functions, as it is presented to us by Legendre, 
may conveniently be considered under the following heads :— 
a. The reduction of the general integral, 
Lf Pdz 
Vat Batya + ox + ext 
in which P is rational to three standard forms, since known as elliptic inte- 
grals of the first, second and third kinds f. 
This classification, though the reduction of the general integral had, as we 
have seen, been already considered by Lagrange, is I believe entirely due to 
Legendre. If we consider how much it has facilitated all subsequent re- 
searches, we can hardly over-rate the importance of the step thus made. | ia 
may almost be said that Legendre, in thus showing us the primary forms with 
which the theory of elliptic integrals is conversant, created a new province 
of analysis: he certainly gave unity and a definite form to the whole sub- 
ject. 
For the three species of functions thus recognised Legendre suggested the 
names of nome, epinome and paranome, the name of the first being derived 
from the idea that it involves, so to speak, the law on which the comparison 
of elliptic integrals depends. But these names do not seem felicitous, nor 
have they I believe been adgpted. To this part of the subject an important 
theorem relating to the reduction of elliptic integrals of the third kind, 
whose parameters are imaginary, seems naturally to belong. 
B. The comparison of elliptic integrals of the same form differing only 
in the value of the variable, or as it is often called, the amplitude of each. 
This part of the subject divides itself into three heads, corresponding to the 
_ three classes of integrals. The fundamental results are to be found in the 
memoirs of Euler, of which we have already spoken. By Legendre how- 
ever they were more fully developed. 
It is interesting to observe that Legendre suggested that the discovery of 
I —- 
__ * A translation of it appeared in Leybourne’s Mathematical Repository, vols. ii, and iii. 
The original I have not seen—it has long been scarce. 
+ These three forms are 
; eae Fy 1—eat, x dz 
af V (1—a®) (=e 2%) SI Set S, Gao 
| a Legendre always replaces 2 by sin @, so that the integrals become 
® ee @ pa 9 ay 
’ SSO nie in2 » SS 
4 , vine or ae he a/1—c? sin? 9d 9; St (1+nsin? 9) /1—c? sin? 
The radical 4/1—c? sin? @ is often denoted by A. 
___ The constant c is called the modulus; the second constant n (in the third kind) is called 
the parameter. The modulus may always be supposed less than unity, and if e=sin s, then 
€ is the angle of the modulus. 
