260 report — 1859. 



probable, when we remember that in the ' Disquisitiones Arithmeticse ' 

 (Art. 335) Gauss promises an "amplum opus" on these transcendents; and 

 that a casual remark of his in relation to them renders it perfectly certain 

 fas Dirichlet has observed)* that he was at that early period in possession 

 of the principle of the double periodicity of elliptic functions — thus antici- 

 pating by twenty-five years the discoveries of Abel and Jacobi. Nevertheless 

 the close analogy we have endeavoured to point out between Gauss's sixtli 

 proof of the quadratic theorem, and the trigonometric demonstration of the 

 biquadratic one, may perhaps incline us to the opposite opinion. Nor is the 

 introduction of complex numbers, as modules, an idea unlikely to have sug- 

 gested itself, when once complex numbers were admitted; though it is 

 remarkable that Jacobi, in the first printed memoir in which complex num- 

 bers appear, and to which we shall presently refer, seems not to have thought 

 of this extension of his theory. 



33. Application of the Lemniscate Functions to the Biquadratic Theorem']'. 

 — Letj3 L be a complex prime (real or imaginary), p its norm; and let the 

 p — \ residues, prime to p v be divided into four groups of \{p— 1) terms, 

 after the following scheme : — 



"S 



(0) r, r 2 r i0 ,_,), 



(0 "'i ir 2 ir i(P-D> 



(2) -r v -r 2 — r i(p -i), 



(3) —ir v —ir 2 —ir kip _ x) , 



so that of any four associated numbers one, and only one, appears in each 

 group. Let q 1 be any residue prime to^; k v k %> k 3 ,... the numbers of the 

 residues 



?i»\ 1x r 2 tfl^CP-D 



which belong to the groups (1), (2), (3), respectively; then 



2 i i(/>-i)= E i*i+2fr3+3fc 3j m od ja^ 



or (1l\ =1*1+5*8+3^. 



(See Gauss, Theor. Res. Biq. Art. 71.) 



The expression on the right-hand side of this equation may now be trans- 

 formed by means of the Lemniscate function <p, defined by the equations 



C * dx 



x=tp(v). 



The function <f>(v~) is doubly periodic, the arguments of the periods being 

 2w and 2«w, or, more properly, (1 +i)io and (1— i)w, where ^=1 —tjz jr; 



so that we have f(v + c 2kw)=(j>(v), k denoting any complex integer what- 

 ever. From this it appears that the relation of the Lemniscate functions to 

 the theory of complex numbers, is the same as the relation of circular func- 



* In his ' Gedachtnissrede iiber Karl Gustav Jacob Jacobi,' Mem. de l'Academie de Berlin, 

 1852. This remarkable eloge is also inserted in Crelle's Journal, vol. Hi., and in a French 

 translation in Liouville's Journal, vol. ii. 2nd series. 



t See Eiseustein's memoir, " Applications de l'Algebre a l'Arithmetique transcendante," in 

 CreUe's Journal, vol. xxx. p. 189, or in Eisenstein's ' Mathematische Abhandlungen,' p. 121. 



