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XXV. Proof of the hitherto undemonstrated Fundamental 

 Theorem of Invariants. By J. J. Sylvester, Professor of 

 Mathematics at the Johns Hopkins University, Baltimore*, 



I AM about to demonstrate a theorem which has been wait- 

 ing proof for the last quarter of a century and upwards. 

 It is the more necessary that this should be done, because the 

 theorem has been supposed to lead to false conclusions, and its 

 correctness has consequently been impugned"]". But, of the two 

 suppositions that might be made to account for the observed 

 discrepancy between the supposed consequences of the theorem 

 and ascertained facts — one that the theorem is false and the 

 reasoning applied to it correct, the other that the theorem is 

 true but that an error was committed in drawing certain de- 

 ductions from it (to which one might add a third, of the 

 theorem and the reasoning upon it being both erroneous). — 

 the wrong alternative was chosen. An error was committed 

 in reasoning out certain supposed consequences of the theorem ; 

 but the theorem itself is perfectly true, as I shall show by an 

 argument so irrefragable that it must be considered for ever 

 hereafter safe from all doubt or cavil. It lies at the basis of 

 the investigations begun by Professor Cayley in his c Second 

 Memoir on Quantics,' which it has fallen to my lot, with no 

 small labour and contention of mind, to lead to a happy issue, 



* Communicated by the Author. 



t Thus in Professor Faa de Bruno's valuable Theorie des Formes 

 Binaires, Turin, 1876, at the foot of page 150 occurs the following pas- 

 sage : — " Oela suppose essentiellement que les equations de condition 

 soient toutes independantes entr'elles, ce qui ri est pas toujour s le cas, ainsi 

 qu'il resulte des recherches du Prof. Gordan sur les nombres des eova- 

 riants des formes quintique et sextique." 



The reader is cautioned against supposing that the consequence alleged 

 above does result from Gordan's researches, which are indubitably correct. 

 This supposed consequence must have arisen from a misapprehension on 

 the part of M. de Bruno of the nature of Professor Cayley's rectification 

 of the error of reasoning contained in his second memoir on Quantics, 

 which had led to results discordant with Gordan's. Thus error breeds 

 error, unless and until the pernicious brood is stamped out for good and all 

 under the iron heel of rigid demonstration. In the early part of this year 

 Mr. Halsted, a Fellow of Johns Hopkins University, called my attention 

 to this passage in M. de Bruno's book ; and all I could say in reply was 

 that " the extrinsic evidence in support of the independence of the equa- 

 tions which had been impugned rendered it in my mind as certain as any 

 fact in nature could be, but that to reduce it to an exact demonstration 

 transcended, I thought, the powers of the human understanding.'' 



At the moment of completing a memoir, to appear in Borchardt's 

 Journal, demonstrating my quarter-of-a-century-old theorem for enabling 

 Invariants to procreate their species, as well by an act of self-fertilization 

 as by conjugation of arbitrarily paired forms, the unhoped and unsought- 

 for prize fell into my lap, and I accomplished with scarcely an effort a task 

 which I had believed lay outside the range of human power. 



