much more easy to cause an electric discharge (without electrodes) to pass through moist oxygen than through the gas when it is dry. The moisture apparently favours the formation of the atoms which carry the discharge. It must be confessed, however, that this view does not explain the formation of ozone which accompanies the oxidation of phosphorus, not only in moist oxygen but also, according to Marchand*, in the dry gas. The complicated nature of the reaction which takes place when phosphorus is oxidized in presence of water makes it impossible to do more than guess at an interpretation of the results obtained. Perhaps, however, the different nature of the equation representing the connexion between the velocity of the reaction and the pressure of the oxygen is due to the water taking part in the reaction. The interesting fact that a pressure of oxygen exists at which the oxidation has a maximum velocity in the case of phosphorus, and perhaps also in that of aldehyde, requires further investigation before any satisfactory attempt can be made to account for it. In conclusion it is perhaps worth noticing that, in one or two other cases which have been studied by other observers, the results are in harmony with the theory of Williamson. Le Châtelier† has shown, using the results of Hautefeuille and Margottet, that, at constant temperature, the equilibrium which occurs when chlorine, hydrogen, and oxygen are exploded together can be represented by the expression in which P(0), P(HCI)... are the partial pressures in the equilibrium of the oxygen, hydrochloric acid,.... If we suppose the reaction to occur between dissociated molecules we may write the reaction which occurs as follows :— 2HC1+0 2C1+H2O, which would correspond to the equation which is the equation given by Le Châtelier after dividing both sides by 2. In conclusion, my best thanks are due to Prof. van't Hoff, in whose laboratory the foregoing work was done, for his advice and assistance during its progress. Journ. prakt. Chem. 1. p. 1 (1850). + Comptes Rendus, cix. p. 665 (1889). LXII. On the Expressibility of a Determinant in Terms of its Coaxial Minors. By THOMAS MUIR, LL.D.* 1. IN a memoir on "A certain Obers; N a memoir on "A certain Class of Generating Functions in the Theory of Numbers," recently published in the Philosophical Transactions†, Major MacMahon, F.R.S., establishes the following noteworthy theorem : In the case of every determinant of even order greater than the second there are two special relations between its coaxial minors, and each of these two relations can be thrown into a form which exhibits the determinant as an irrational function of its coaxial minors: in the case of a determinant of odd order, on the other hand, no such relations exist, and it is not possible to express the determinant as a function of its coaxial minors. He deduces the theorem readily from another to the effect that There are 2"-n2+n-2 relations between the coaxial minors of any determinant of the nth order. His proof of this latter theorem, however, is not by any means simple, occupying as many as eight pages (pp. 133-140) of the memoir. By reason of the importance of the theorem a simpler proof is much to be desired, and part of my object at present is to supply the want. 2. I start from the familiar proposition, that if the rows of a determinant of the nth order be multiplied by X1, X2, X3........., Xn respectively, and the columns be then divided by the same quantities, the determinant is unaltered in value; but I prefer to include it in a more general but equally evident theorem, viz. : If the rows of a determinant of the nth order be multiplied by X1, X2, X3,..., Xn respectively, and the columns be then divided by X1, X2, X3,..., Xn respectively, the determinant is unaltered in value, and each of the minors of the transformed determinant is, to a factor près, equal to the corresponding minor of the original determinant, the connecting multiplier being XXX.../X, X, Xt... if the minor belong to the hth, kth, 1th, ",... rows, and rth, sth, tth, columns of the original. From this we have manifestly the corollary: The connecting multiplier in the case of the coaxial minors, as in the case of the whole determinant, is 1: in other words, the coaxial minors remain unaltered by the transformation. *Communicated by the Author. † Vol. clxxxv. (1894) pp. 111–160. Phil. Mag. S. 5. Vol. 38. No. 235. Dec. 1894. 20 Next it is clear that X1, X2, X3, .., Xn may be so chosen that all the elements of any one of the rows or columns, except the diagonal element, shall be 1. Any determinant of the nth order may be transformed so as to have 1 for n-1 of its elements, and yet the determinant itself and all its coaxial minors remain unaltered in value. 3. This is the same as saying that 2"-1 quantities, viz. the determinant and its coaxial minors, can be expressed in terms of n2-(n-1) others, viz. the modified elements, which are not equal to unity. Eliminating the latter, and we have 2-n2+n-2 relations connecting the former-and this is Major MacMahon's auxiliary theorem. 4. The proof leaves no doubt as to the existence of two relations between any determinant, when of even order higher than the second, and its coaxial minors: but it is worth while to intensify the conviction by putting the relations actually in evidence for a particular determinant. Perhaps the determinant which lends itself most easily to the end in view is Here the four coaxial minors of the 3rd order are { Calling these A, B, C, D respectively, and denoting the original determinant by A, we have the five relations Clearly from these a, b, c may be eliminated, and five relations found, viz. connecting ABCD, ABCA, ABDA, ACDA, BCDA: manifestly, however, only two of the relations can be independent. 5. To find the simplest of the five, viz. that which connects ABCD, let us call the two values of a in the first equation a and, the two values of b in the second equation and 1 the two values of c in the third equation y and 1 B B' and let us substitute these values in the fourth equation. The elimi nant is thus seen to be S B αγ α -(2+D) + В - (2 + D) + a3y } { ay By By} LaB aß =0; aß {D+ − − αβ γ - (2 + D) + αβ γ aßy or (D+2) - (D+2)3 { α + By B ya ya ав + + + +aßy+ +(D+2)3 { +++By+ya2+a"/8° 1 y2a2+a2ß2 ++++++ βγ α ya B +Σ + Σ + Σα βγ 1 } γ +23 + 2 2 + ਕ +1} = 0. a2ß" a2ß2y2 By utilizing the facts that a+1=A+2, 8+ 1 = B+2, y+1=0+2, a this is readily transformed into (D+2)+ −(D+2)3(A+2)(B+2)(C+2) +(D+2)2 { Σ(A+ 2)2(B + 2)2 − 2Σ(A+2)2} αβγ −(D+2) {Σ(A+2)3(B+2)(C+2) −8(A+2)(B+2)(C+2)} + {Σ(A+2)1-2Σ(A+2)2(B+2)2 +(A+2)(B+2)2(C+2)2}=0; and by applying the Σ to the four letters A, B, C, D, it becomes Σ(A+2)1+ Σ(A + 2)2 (B+ 2)2(C+2)2 + 8 (A+2) (B+2) (C+2) (D+2| −Σ (A+2)3 (B+2) (C +2) (D + 2) −2Σ(A+ 2)2(B+ 2)2=0. |