Sylvester's and other Unisignants, 
189 
instead of being 0 as in the case of or even being known to be 
unisignant, requires special investigation. 
This, therefore, we proceed to give, and in doing so will for shortness' 
sake write 
'^234, ^12. ••• ^ov ^2 + ^3 + ^4, 61 + ^2, ... 
and denote the determinant itself by 2 
5. At the outset we note that much may be learned regarding its 
constitution by making transpositions of rows followed by the like 
transpositions of columns. Thus, by passing the first row over all the 
others, and thereafter moving the first column in the same way, it is seen 
that the determinant is invariant to the triad of cyclic substitutions 
'^2» ^^3> d, = ^3, C4, ^2 ) 
^3, h\, c„ d^ = 64, c„ 6^3 V 
a^, b„ C2, d^ - b„ C2, ^^3, 1 
or, what is the same thing, the pair 
a, h, c, d = h, c, d, a \ 
1, 2, 3, 4 = 2, 3, 4, 1 j 
(III.) 
Again, by passing the second row over the remaining rows and thereafter 
moving the second column in the same way, we deduce the, existence of 
invariance to the quartet of cyclic substitutions 
^3, = a^, a^, 
61, Ci, d^ = Ci, d^, 
63, C4, C?2 ~ <^4> ^2> ^3 
64. C2, ^3 = C^, rfj, 
(IV. 
Both of these results have their advantages as assisting in finding the 
final development and as affording a means of abbreviating the expres- 
sions for the same. 
6. We then seek a form of more suitable for development, and 
obtain 
<X234 
^2 
a. 
a, 
a. 
^341 
^341 
-K 
^34x 
^3 
K 
C4X2 
C2 
C 
412 
^4 
-C24 
C2 
C4X2 
f^I23 
d 
3 
f/123 
-^23 
6^2 
^3 
di2^ 
a 
-K 
-^3 
^24 
■ ^4 
- c. 
- d^ 
-d. 
d. 
(V.) 
