ADDRESS. 29 
rise to some very interesting developments—and omitting to mention at all 
various functions of minor importance,—we come (1811-1829) to the very 
wide groups, the elliptic functions and the single theta-functions. I give 
the interval of date so as to include Legendre’s two systematic works, the 
‘Exercises de Calcul Intégral’ (1811-1816) and the ‘ Théorie des Fonctions 
Elliptiques ’ (1825-1828); also Jacobi’s ‘ Fundamenta nova theorize Func- 
_ tionum Ellipticarum ’ (1829), calling to mind that many of Jacobi’s results 
were obtained simultaneously by Abel. I remark that Legendre started 
from the consideration of the integrals depending on a radical ./ X, the 
square root of a rational and integral quartic function of a variable «; for 
this he substituted a radical Ad, = ./1—k*sin?¢, and he arrived e his 
three kinds of elliptic integrals Fy, H¢, If, depending on the argument 
or amplitude 9, the modulus /, and also the last of them on a parameter n ; 
the function F is properly an inverse function, and in place of it Abel and 
Jacobi each of them introduced the direct functions corresponding to 
the circular functions sine and cosine, Abel’s functions called by him 
o,f, F, and Jacobi’s functions sinam, cosam, Aam, or as they are also 
written sn, cn, dn. Jacobi, moreover, in the development of his theory of 
transformation obtained a multitude of formule containing q, a tran- 
scendental function of the modulus defined by the equation g=e—"*", 
and he was also led by it to consider the two new functions H, ®, which 
(taken each separately with two different arguments) are in fact the 
four functions called elsewhere by him @,, ©,, ©, @,; these are the 
so-called theta-functions, or, when the distinction is necessary, the single 
theta-functions. Finally, Jacobi using tke transformation sin ¢=sinam u, 
expressed Legendre’s integral of the second and third kinds as integrals 
depending on the new variable u, denoting them by means of the letters 
Z, Ul, and connecting them with his own functions H and @: and the 
elliptic functions sn, cn, dn are expressed with these, or say with 
@,, 95, ©3. ©,, as fractions having a common denominator. 
Tt may be convenient to mention that Hermite in 1858, introducing 
into the theory in place of g the new variable w connected with it by the 
equation =e’ (so that w is in fact = 7K’ /K), was led to consider the three 
functions ¢w, yw, xv, which denote respectively the values of 4/k, 4/// 
and 1%/kk’ regarded as functions of w. A theta-function, putting the 
argument = 0, and then regarding itas a function of », is what Professor 
Smith in a valuable memoir, left incomplete by his dean: calls an omega- 
function, and the three functions ¢w, Ww, xw are his modular functions. 
The proper elliptic functions sn, en, dn form a system very analogous 
to the circular functions sine and cosine (say they are a sine and two 
separate cosines), having a like addition-theorem, viz. the form of this 
theorem is that the sn, cn and dn of #+y are each of them ex- 
pressible rationally in terms of the sn, cn and dn of x and of the sn, 
en and dn of y; and in fact reducing itself to the system of the 
circular functions in the particular case /=0. But there is the 
important difference of form that the expressions for the sn, cn and 
