451 
1910-11.] Dr Muir on Boole’s Unisignant. 
In other words, the first and last variables being retained, it is allow- 
able to take any 'permutation of the first triad, provided the same 
permutation of the second triad be taken at the same time. (IV.) 
6. Again, by subtracting the first row from one or more of the other 
rows, and subsequently treating the columns in the same way, we obtain 
B(6, agf, hdc, e) , \ 
B (c, gae, dhb, f), - 
B (d , f e a, c b h , g) , 
B(e , hdc , agf, b),\ 
B (/, dhb, gae, c) , V 
B (g , cbh , f e a, d ) , ) 
B (h , e f g , bed , a) . 
In other words, we learn that — 
The first letter may be interchanged with any one of the letters of the 
first triad, provided that the last letter be interchanged with the corre- 
sponding letter of the second triad, and the remaining letters of the 
first triad be interchanged with the corresponding letters of the second 
triad. (Y.) 
The first letter may be interchanged with any one of the letters of the 
second triad, provided the last letter be interchanged with the correspond- 
ing letter of the first triad. (YI.) 
The first and last letters may be interchanged, provided all the letters 
of the one triad be interchanged in order with the corresponding letters 
of the other. (YII.) 
7. From a combination of the results of §§ 5, 6 it is seen that there 
are in all forty-eight different orders in which the variables can be taken. 
If we note that from (IY.) and (YII.) the order of the variables may be 
reversed, we may obtain the forty-eight forms most readily by using (Y.) 
or (YI.), then performing reversal, and finally using (IY.). 
As one of the results of § 5 is to the effect that the function is invariant 
to the simultaneous performance of the cyclical substitutions 
b , c , d = c , d,b \ 
= f,g,ei 
we may, in general, on knowing one term of the development, obtain in 
this way two others. Thus ecdh gives rise to fdbh and gbch, and we may 
also, therefore, conveniently denote the sum of the three by 
2 ecdh . 
