324 
Transactions of the Boyal Society of South Africa. 
The original proof is neither short nor simple, the method followed being 
that known as "mathematical induction." From this some important 
conclusions are drawn, not the least interesting being those which are 
shown to link themselves on to a special class of determinants studied 
long before by Sylvester, and styled by him " inversely orthogonal 
determinants." * 
3. In 1902 the subject acquired a still greater importance because of 
the intimate connection which it was found to have with Fredholm's 
equation.! This brought Professor Wirtinger to bestow attention on it, 
with the result that in 1907 he published | a fresh proof of the funda- 
mental theorem, his mode of procedure being to apply the ordinary 
Lagrangian rule for finding by differentiation the extreme values of a 
function whose variables are connected by equations of condition. This 
proof, though claimed to be shorter, and though having, of course, its own 
points of interest, cannot be said to be essentially simpler than that of 
Hadamard. 
4. In these circumstances it seems desirable to point out, as I now 
propose to do, that my original method of treating the special case of the 
theorem is equally applicable when the elements of the determinant are 
complex quantities. Further, this having been done, it will readily appear 
that a fresh and simple presentment of the whole subject follows there- 
from in a very natural way. 
5. Denoting the determinant 
and its adjugate by |Aj + A'i Ba + B',?' G. + G\i\ or M, we have for 
fi = {a, + a>')(A, + A;i) + {h, + h\.i){B, + B;i) + {c, + c\:i){G, + G',i) 
= (a,A, - a',A', + b,B, - b',B', + c,G, - c',C\) + (a,A', + a,A, + b,B', + KB, 
* Sylvester, J. J., Thoughts on inverse orthogonal matrices, . . . Pliilos. Magazine 
(4), xxxiv. pp. 461-475 ; or Collected Math. Papers, ii., pp. 615-628. 
f Fredholm, I., Sur una classe de transformations rationnelles. Gomptes-rendus . . . 
Acad, des Sci. (Paris), cxxxiv., pp. 219-222, 1561-1564. 
Fredholm, I., Sur une classe d'cquations fonctionnelles. Acta Math., xxvii., 
pp. 365-390. 
I Wirtinger, W., Zum Hadamardschen Determinantensatz. Monatshefte f. Math, 
u. Phys., xviii., pp. 158-160 ; or Bull, des Sci. math. (2), xxxi., pp. 175-179. 
ft I + a,i a 2 + a A a^ + a'^i : 
hi + h\i b^ + b'.i b.+ b\i\ by ^ 
Ci + c\i C2 + c[i <? 3 + c\i 
r = l, 2, 3, 
= (^a^.A, - Za[A',) + (Sa,.A;. + 2a', A,.) i. 
+ c,.C:. + c,C,)i 
(!•) 
