158 
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION 
[662 
viz. we have P 2 = aX' — /3F' — <yZ' + 8W', etc. The reason for the apparently arbitrary 
manner in which I have numbered these equations, will appear further on. I recall 
that the sixteen double ©-functions, that is, ©-functions of two arguments u, u, are* 
P, P x , P 2 , P 3 , 
Qi> iQs> Qs> 
%R, tPj, P'S) 
S, iS±, iS 2 , S 3 , 
the factor i, = V— 1, being introduced in regard to the six functions which are odd 
functions of the arguments. But disregarding the sign, I speak of P 2 , P 2 , ... , Q 2 , etc., 
as the squared functions, or simply as the squares ; a, /3, 7, S are constants, depending 
of course on the parameters of the ©-functions ; X', Y', Z', W', which are however 
to be eliminated, are themselves ©-functions to a different set of parameters : the 
16 equations express that the squared functions P 2 , P 2 , etc., are linear functions of 
X', Y', Z', W, and they consequently serve to obtain linear relations between the 
squared functions: viz. by means of them, Gopel expresses the remaining 12 squares, 
each in terms of the selected four squares P 2 , P 2 2 , S 2 , S 2 2 , which are linearly inde 
pendent : that is, he obtains linear relations between five squares, and he seems to 
have assumed that there were not any linear relations between fewer than five squares. 
It appears however by Rosenhain’s “Mémoire sur les fonctions de deux variables 
et à quatre périodes etc.”, Mém. Sav. Étrangers, t. xi. (1851), pp. .864—468, that 
there are, in fact, linear relations between four squares, viz. that there exist sixes of 
squares such that, selecting at pleasure any three out of the six, each of the 
remaining three squares can be expressed as a linear function of these three squares. 
Knowing this result, it is easy to verify it by means of the sixteen equations, and 
moreover to show that there are in all 16 such sixes: these are shown by the following- 
scheme which I copy from Rummer’s memoir “ Ueber die algebraischen Strahlensysteme 
u. s. w.” Berlin. Abh. (1866), p. 66 : viz. the scheme is 
1 
2 
8 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
9 
10 
11 
12 
13 
14 
15 
16 
1 
2 
O 
O 
4 
5 
6 
7 
8 
13 
14 
15 
16 
9 
10 
11 
12 
5 
6 
7 
8 
1 
2 
3 
4 
8 
7 
6 
5 
4 
3 
2 
1 
16 
15 
14 
13 
12 
11 
10 
9 
7 
8 
5 
6 
3 
4 
1 
2 
15 
16 
13 
14 
11 
12 
9 
10 
6 
5 
8 
7 
2 
1 
4 
3 
14 
13 
16 
15 
10 
9 
12 
11. 
* The same functions in Bosenhain’s notation are 
00, 02, 20, 22, 
01, 03, 21, 23, 
10, 12, 30, 32, 
11, 13, 31, 33; 
viz. the figures here written down are the suffixes of his ^--functions, 00=3- 0)0 (v, iv), etc.