194 Transactions of the Boyal Society of South Africa. 
minors, each of which is seen to be an S3. The terms are thus all 
negative, the number being 
4 + 6.8 + 4.48 + 4.4.16, 
i.e., 500. 
In like manner the case of S5 may next be established, then 85, S7, .... 
13. The theorem just proved may also be stated as follows : The sum 
of the signed primary minors of a Sylvester's unisignant is also unisignant. 
On putting all the variables equal to 1 in the bordered S„, it is readily 
seen that the number of terms in it is n times the number in S„ itself, and 
therefore is 
n{n + lf-\ (XV.) 
14. Careful note should be taken of the fact that the like bordering of 
any other unisignant will not necessarily be attended with the same 
result. Thus in the case of the unisignant which most closely resembles 
S,„ namely T„, the property does not hold. It holds, however, in the 
case of Boole's unisignant B,,, and in the case of the " new unisignant " 
M„ recently described by me in the Messenger of Math., xl., pp. 177-192. 
The last of these cases is by far the most interesting, as the bordered 
imisignant is then transformable into a like unisignant of the next loiver 
order, the exact result for the third order being that the sum of the signed 
primary minors of 
M(«i, bJ)J)y Cj^c^c^ 
is 
M(6, + c,, b^ + c^, 63 + C3). (XVI.) 
In order to make clear the mode of establishing this, let us take the 
more complicated case of M^, where the sum of the signed primary 
minors is 
- . 1 1 1 1 
1 a + 6234 + C123 + I (X + 634 + C1 a + Z)24 + C2 a + ^23 + 03 
1 a + b^^ + c^ ^^^ + &i34 + Ci45 + r/2 a + b^^^c^ a + b^^->rC^ 
1 a + Z>24 + C2 a + b^^+c^ a + Z>i2_^ + C246 + (^3 a^b^^-\-Ce, 
1 a + b^^ + c-. a + b^^ + c^ a-\-b^^ + Ce a + b^^^ + c^^^-^d^ . 
The operations which transform this into 
- . 1 1 1 1 
1 Ci23 + f?i c^ — b^ c^ — b^ ^3 — ^1 
1 c^-b^ Ci45 + f/2 c^-b^ 05-l>2 
1 ^2-^3 c^-b^ c^^e + d^ ^6-^3 
1 C3-64 Co-b^ ^356 + 0(4 
