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AN UPPEE LIMIT FOE THE VALUE OF A DETEEMINANT. 
By Thomas Mum, LL.D., F.E.S. 
(Eead March 17, 1909.) 
1. In 1885 I was led to establish the theorem, If he the sum of the 
squares of the elements of the rth row of a determinant h, and S,'. be the sum 
of the squares of the elements of the corresponding roio of the adjugate 
determinant A, then Z < s,.S,. : and by means of it succeeded in proving for 
Sir William Thomson (afterward Lord Kelvin) the fm^ther inequality 
^ = S1S2S3 ... 
Taking, for example, the case where c is of the fourth order, we obtain by 
using the former theorem four times 
and therefore by Cauchy's theorem regarding the adjugate determinant 
S1S2S3S4.S1S2S3S4 > 
From this the second theorem results at once. 
Although it was agreed at the time that the latter theorem should be 
published in the Educational Times, it did not actually appear until 1901.* 
2. It was in 1893 that the subject first assumed importance, M. 
Hadamard having in that year drawn the attention of mathematicians to 
it by means of two different papers.! His fundamental result, which is 
an extension of the theorem just mentioned so as to include determinants 
with complex elements, may be formally enunciated thus : If s,-. he the sum 
* See Educ. Times, liv., p. 83, or Math, from Educ. Times (2), i., pp. 52, 53. The 
date of the theorem was there given from memory as being 1886. It should have been 
1885. Lord Kelvin's letter approving of publication and remarking on the proof has since 
been recovered, and is dated " Nov. 12/85." [This letter has been duly shown to me as 
President of the Society. — S. S. Hough.] 
f Hadamard, J., Eesolution d'une question relative aux determinants. Bull, des Sci. 
math. (2), xvii., pp. 240-246. 
Hadamard, J., Sur le module maximum que puisse atteindre un determinant. 
Comptes-rendus , . . Acad, des Sci. (Paris), cxi,, pp. 1500-1501. 
