362 
Proceedings of the Royal Society of Edinburgh. 
[Sess. 
and 
cos 2 ^r 
108 
7 r • 9 Lq 
7 
, 2 , sin 2 4§- 
\ + 5 cos toF^ -i- , 
5 
, 2 , sin 2 -^) 
2 
L V 6 
1 6 
2, 
1 3 \6 ' 
' 6 
2 / J 
+ 
VII. 
As there is an infinite number of cubic equations with real roots, it is 
impossible to tabulate the periods of all possible elliptic functions with 
either the roots or the invariants as arguments. The formulse of $ 5 show 
that the periods depend on the absolute invariant q and the quantity X v a 
simple function of the invariant g 2 . 
For the purposes of tabulation we may assume that we can find a factor 
which will change e x into unity or a factor which will similarly reduce to 
unity. Let us assume that we have found a factor m 2 which will reduce e 1 
to unity. Whatever m may be, the following relations hold : — 
m 2 e 1 + m 2 e 2 + m 2 e 3 = 0 
m 2 e 1 m 2 e 2 + m 2 e 1 m 2 e 3 + m 2 e 2 ra 2 e 3 = - Jm 4 # 2 
etc. 
>(S 1 5 
m4?! ’ m ° g *) g 
This assumption will enable us to form a table which will include within 
its limits every possible case of three real roots. Such a table follows. 
As there is a close relationship between the forms used by Legendre and 
Jacobi and those of Weierstrass, I have included in the table the values of 
Legendre’s modular angle 0 and h and /q , and Jacobi’s nome q v 
The relation between Legendre’s modular angle and the absolute 
invariant i is : — 
2 tan h- 
sin 2 0 = — • 
J 3 + tan h- 
The period n x vanishes for log q x = 9‘2857. 
