170 
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
[663 
column: and it appears that there are four columns in each of which the signs are 
or can be made all +; whereas in each of the remaining twelve columns the signs 
are or can be made six of them +, the other four —. 
Passing to the values of {a}, [ah], etc., we have for example, from the ab column 
of the foregoing table, 
{c} = + Vc. etc. be, 
{*2} = + fd. ad. bd, 
|ac} = — Vac. 
ac. ae 
bc.bf’ 
where (since the radicals are all positive) the signs are correct: substituting for the 
quantities under the radical signs their full values, and squaring the rational parts in 
order to bring them also under the radical signs, this is 
{c} = 4- fab . ad . ae . af. bd .be . bf. de . df. ef. ac 2 . be 2 , 
[d\ = + fab.ac. ae. af. bc.be . bf. ce .cf. ef. ad 2 . bd 2 , 
[ac} = — Vac . af. cf. bd .be.de. ac 2 . ae 2 . be 2 . bf 2 , 
where all the expressions of this (the afr-column) have a common factor, 
ac . ad. ae. af. be .bd .be. bf 
Omitting this factor, we find 
{c} = + V ab .ac .be . de. df. ef, 
\d] = + fob . ad . bd . ce. cf. ef, 
{ac} = — Vad . ae . de .be . bf. cf; 
viz. recurring to the foregoing condensed notation, this is 
{c} = + V de, 
{¿} = + fee, 
{oc} = — fbc, 
and, in fact, the terms in the several columns have only the ten values fab, fac, 
etc. each with its proper sign. I repeat the meaning of the notation: ab stands in 
the first instance for the double triad abf. ede, and then this denotes a product of 
differences ab.af.bf.cd.ce.de. We have thus the following table in which I have 
in several cases changed the signs of entire columns.