142 Mr. J. J. Sylvester on a certain 



steam is not so completely compensated by heat as in the first 

 case, I could notwithstanding affirm in my investigation, in 

 which it was not my design specially to discuss all incidental 

 circumstances, but which was directed to the establishment of a 

 principle, that the widening neck was not necessary to the va- 

 lidity of the reasoning. 



I believe I have thus justified the views to which I have given 

 utterance. 



With regard to the friction of the steam as it rushes through 

 the orifice, I have arrived at the conclusion that it is not neces- 

 sary to the explanation of the fact adduced by Mr. Thomson. 

 At the same time, its action, which, according to my view, would 

 be the reverse of that imagined by Mr. Thomson, and must be 

 introduced as a loss of lieat into the calculation, is by no means 

 excluded. In the case of a very small orifice, its influence may 

 be even considerable. If, however, in such cases as the issuing 

 of the steam through the safety-valve of a high-pressure engine 

 I have regarded it as playing a less important part than that 

 attributed to it by Mr. Thomson, this opinion will not be con- 

 sidered as groundless by those who understand the subject. 



XXIV. On a certain Fundamental Theorem of Determinants, 

 By J. J. Sylvester, M.A., Dub,^ 



THE subjoined theorem, w^hich is one susceptible of great 

 extension and generalization, appears to me, and indeed 

 from use and acquaintance (it having been long in my posses- 

 sion) I know to be so important and fundamental, as to induce 

 me to extract it from a mass of memoranda on the same subject ; 

 and as an act of duty to my fellow -labourers in the theory of de- 

 terminants, more or less forestall time (the sure discoverer of 

 all truth) by placing it without further delay on record in the 

 pages of this Magazine. Its developments and applications 

 must be reserved for a more convenient occasion, when the in- 

 terest in the New Algebra (for such, truly, it is the office of the 

 theory of determinants to establish), and the number of its dis- 

 ciples in this country, shall have received its destined augmen- 

 tation. In a recent letter to me, M. Hermite well alludes to the 

 theoi-y of determinants as " That vast theory, transcendental in 

 point of difficulty, elementaiy in regard to its being the basis of 

 researches in the higher arithmetic and in analytical geometry." 

 The theorem is as follows : — Suppose that there are two de- 

 terminants of the ordinary kind, each expressed by a square 

 array of terms made up of n lines and n columns, so that in each 



* Communicated by the Author. 



