164 
ON THE DOUBLE ©-FUNCTIONS IN CONNEXION 
[662 
(12) AD = y——;- 2 {Cbcea'df — Vb'c'e'adf } 2 , 
(11) AC — t—— {Vbdea'cf - Cb'd'e'acf}' 2 , 
[CC oc y 
(5) AB = ——7- {Ccdea'bf' — Ccd'e'abf} 2 , 
yCO co y 
where the numbers are in accordance with the foregoing scheme; viz. the scheme 
becomes 
(1) 
(2) 
(3) 
(4) 
(5) 
(6) 
a) 
(8) 
(9) 
(10) 
(H) 
(12) 
(13) 
(14) 
(15) 
(16) 
F 
CD 
DE 
CE 
AB 
E 
c 
D 
B 
AE 
AC 
AD 
A 
BE 
BC 
BD 
B 
AE 
AC 
AD 
A 
BE 
BC 
BD 
F 
CD 
DE 
CE 
AB 
E 
C 
D 
A 
BE 
BC 
BD 
B 
AE 
AC 
AD 
AB 
E 
C 
D 
F 
CD 
DE 
CE 
D 
C 
E 
AB 
CE 
DE 
CD 
F 
BD 
BC 
BE 
A 
AD 
AC 
AE 
B 
C 
D 
AB 
E 
DE 
CE 
F 
CD 
BC 
BD 
A 
BE 
AC 
AD 
B 
AE 
E 
AB 
D 
C 
CD 
F 
CE 
DE 
DE 
A 
BD 
BC 
AE 
B 
AD 
AC. 
There is 
of course 
the six A, 
B, C, D, 
E, F; for 
each 
of these is a ] 
linear 
function of 1, x + x', xx, and there is thus a linear relation between any four of 
them. It would at first sight appear that the remaining sixes were of two different 
forms, A, B, AB, CE, CD, DE, and F, A, AB, AC, AD, AE; but these are really 
identical, for taking any two letters E, F, the six is E, F, AE, BE, CE, DE, or, as 
this might be written, E, F, AEF, BEF, CEF, DEF, where AEF means BCD . AEF, 
etc.; and we thus obtain each of the remaining fifteen sixes. The six just referred 
to, viz. E, F, AE, BE, CE, DE, or changing the notation say E, F, A, B, C, D as 
indicated in the table, thus represents any one of the sixes other than the rational 
six A, B, C, D, E, F; and there is no difficulty in actually finding each of the fifteen 
relations between four functions of the six in question, E, F, A, B, C, D. It is to 
be observed that every such function as A contains the same irrational part 
.———7— Oabcdefa'b'c'd'e'f, 
[CO CC ) 
and that the linear relations involve therefore only the differences A — B, A — C, etc., 
which are rational. Proceeding to calculate these differences, we have for instance 
C — D = .——tt 2 (cefa'b'd' + c'e'fabd — defa’b'c — d'ef'abc) = j——(cd' - c'd)(efab' — e'fab); 
(co — cc j \C0 x y 
or, substituting for a, a’, etc. their values a — x, a — x', etc., we have 
cd' — c'd = (x — x) (c — d), 
efa'b' — e'fab = {x — x) 
1, x 4- x, 
1, a+b, 
1> e +/> 
xx 
ab 
ef 
= {x — x) \xx'abef\ 
or say for shortness