168 
FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
[663 
double triads for the four functions are, in fact, abf.cde, cdf.abe, cef.abd, def.abc)\ 
but the other six functions [ab], for which the letters a, b occur in separate triads, 
acquire determinate values. 
It is important to attend to the signs : for example, if x, x = b, e, we have 
[c] = ce. cb, = — bc. ce 
P , _ 1 cb .eb .fb _ cb .fb 
Ce (be) 2 ae.be .de’ ae.de’ 
be. bf 
ae .de' 
Table I. of the values of [a], [a&], etc., 
æ, 
x' = CO 00 
ab 
ac 
ad 
ae 
af 
be 
bd 
M 
0 
0 
0 
0 
0 
0 
+ ab. ac 
+ ab.ad 
[6] 
0 
0 
— ab .be 
— ab . bd 
— ab .be 
— ab .bf 
0 
0 
M 
0 
+ ac .be 
0 
— ac . cd 
— ac .ce 
— ac .cf 
0 
— be . cd 
M 
0 
+ ad.bd 
+ ad. cd 
0 
— ad .de 
— ad .df 
+ bd. cd 
0 
M 
0 
+ ae .be 
+ ae . ce 
+ ae . de 
0 
— ae .ef 
+ be . ce 
+ be .de 
[/] 
0 
+ a/, b/ 
+ af. cf 
+ »/• df 
+ af .ef 
0 
+ ¥■ cf 
+ ¥■ df 
[a&] 
+ ab/, ede 
0 
ad. ae 
+ be. cf 
ac . ae 
+ bd.d/ 
ac. ad 
+ be. ef 
0 
ac. bd 
+ be .cf 
ad. be 
+ be .df 
[ac] 
+ acf. bde 
ad. ae 
be . b/ 
0 
ab . ae 
+ cd .cf 
ab. ad 
+ ce .ef 
0 
ab. bf 
cd. ce 
0 
[ad] 
+ ad/, bee 
ac. ae 
bd. b/ 
ab . ae 
cd. c/ 
0 
ab . ac 
+ de. ef 
0 
0 
ab. bf 
cd. de 
[ae] 
+ aef. bed 
ac. ae 
~ be. b/ 
ab . ad 
ce . cf 
ab. ac 
de. d/ 
0 
0 
0 
0 
[be] 
+ be/. ade 
ac. a/ 
bd. be 
ab . a/ 
cd. ce 
0 
0 
ab . ac 
~ df.ef 
0 
ab. be 
cd. df 
[bd] 
+ bd/. ace 
ad. a/ 
be . be 
0 
ab. af 
+ J 
cd. de 
0 
ab . ad 
~ <if .ef 
ab . be 
+ cd .cf 
0 
[be] 
+ be/. acd 
ae. a/ 
be . bd 
0 
0 
ab . af 
ce . de 
ab . ae 
cf .df 
ab . bd 
+ r 
ce .cf 
ab. be 
+ de .df 
[cd] 
+ cd/. abe 
0 
ad. af 
+ be .ce 
ac. af 
bd. de 
0 
ac . ad 
¥ -cf 
ac. bd 
+ bf. ce 
ad. be 
+ bf .de 
[ce] 
+ ce/. abd 
0 
ae . a/ 
be . cd 
0 
ac . af 
be . de 
ac. ae 
~bf.df 
ac . be 
+ bf .cd 
0 
[i/e] 
+ de/. abc 
0 
0 
ae. a/ 
be. cd 
ad. af 
be . ce 
ad. ae 
¥■ cf 
0 
ad. be 
bf. cd