206 
Transactions of the Boyal Society of South Africa. 
we have, of course 
ff = a^A^ + a^A^ + a^A^ + a^A. + a^A^ 
ff = a,A, + 53B3 + hfi, ^r h,^, + he^e 
ff - ^3 A3 + ^^363 + + C5C5 + CeGe 
/ = a,A, + hfi^ + c,0, + dp, + ^^gDe [" 
/ - a- A, + &5B5 + + rZjD^ + ggEg 
/ = aeAe + ^6^6 + c^Ge + 4I>6 + ^6^5 ) , 
where the oblong assemblage of terms on the right has a peculiar 
approximation to axisymmetry and can be made actually axisymmetric 
by introducing a zero diagonal. Subtracting now any one of the six 
expressions, say the first, from the sum of the remaining five we obtain 
and , 
4/= 2(63B + 6,B, + ...+«,E,), 
.2/= (63 b, b.hlB, B,B3 B, 
C4 C5 C| 
and five others similar to it. Again, by subtracting the sum of any two, 
say the first and second, from the sum of the remaining four, and dividing 
by 2 we find 
/= (c, c, Ce\G, C5 Cg) - a, A3, 
d. de 
D5 
and fourteen others similar to it. And, lastly, by subtracting the sum of 
any three from the sum of the remaining three we obtain 
0 = {d, deWs ^e) - {CL, A3) 
e6\ Eel h,\ B3I 
and nineteen others similar to it. Among the nineteen is the identity 
with which we started. 
The general theorem may be enunciated thus: If /z, v he complementary 
minors of a Pfaffian, and M, N the corresponding minors of the adjugate, 
then 
(^?M)-(v$N) 
is a multiple of the Pfajflan^ the midtiplier being 0 ivhen fx, v are of the 
same order. 
3. In determinants there is an almost perfectly analogous theorem. 
Here, however, instead of having one set of expressions, all of which are 
