52 REPORT—1846. 
from the algebraical integrals connected with them, and therefore these in- 
tegrals are not brought so much into view as in the theory of elliptic func- 
tions the corresponding integrals necessarily are; but it is certain that while 
the discrepancy continued to exist the subject could never be fully or satis- 
factorily developed. The maxim “ verba vestigia mentis” is as true of ma- 
thematical symbols as of the elements of ordinary language. 4 
We shall see hereafter that Abel took the same step in his first essay on 
elliptic functions. At present I shall only remark, that one of the earliest 
consequences of the new notation was the recognition of a most important 
principle, viz. that the “inverse function” sinam uw, that is, the function 
q 
f 
corresponding to sin x in circular functions, is doubly periodic, or that it re- 
tains the same value when w increases by any multiple either of a certain 
real or of a certain imaginary quantity. Now M. Jacobi has shown that no 
function* can be triply periodic, and therefore these inverse functions pos- 
sess the most general kind possible of periodicity, a property which gives 
them great analytical importance. , 
Following M. Jacobi, we shall henceforth give the name of elliptic func- 
tions to those which are analogous to circular functions. It is on this ac- 
count better to call Legendre’s functions elliptic integrals than, as he has 
done, elliptic functions (vide ante, p. 44). 
By the new notation we are led to consider a great variety of formule 
analogous to those of ordinary trigonometry. The sine or cosine of the am- 
plitude of the sum of two quantities may be expressed in terms of the sines 
and cosines of the amplitudes of each, &c.+; and we have only to make the 
modulus equal to zero to pass from what has sometimes, though not with 
much propriety, been called elliptic trigonometry to the common properties 
of circular functions. ; 
M. Jacobi gives a table of formule relating to the new elliptic functions, 
and proceeds to apply their properties to the problem of transformation. It 
was in this manner that he had treated the problem in the Nachrichten. As 
* 7, e, no function of one variable. 
+ The fundamental formule are— 
sinam ucosamv Aamv- sinam v cosamu A amu 
1 — #* sin? am wu sin? am v : 
sin am (w+ v) = 
cos am ucosamv — sinam usinamvA amu Aamv 
] — #* sin? am u sin? am v 3 
cos am (wu + v) = 
AamuAamv — /* sin am uw sin am v Cos am wu cos am » 
Aam (u + v) = 1 — # sin? am u sin? am v 4 
k being the modulus, and Aamu= V1—A*sin?amu. If 
* = 
K= : pee 5 Me and K’ = Sih 
» Vi-#sin? 9 » V1—k?* sin? 
where #? + #/2 = 1, then it may be shown that 
sin am (u-+ 4K) = sinam uw, 
and cham 
sin am (wu + 2K’ /—1) = sinam u, 
so that 4 K is the real and 2K’ /—1 the imaginary period of sinamwu. Hence it is ob- 
vious that we shall have generally 
sin am (u + 4mK-+ 2nK’ /—1) =sinam u, 
m and n being any integers. 
