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 waiting 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 impugnedt. 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 deductions 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 '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. 66 Thus in Professor Faà de Bruno's valuable Théorie des Formes Binaires, Turin, 1876, at the foot of page 150 occurs the following passage::- Cela suppose essentiellement que les équations de condition soient toutes indépendantes entr'elles, ce qui n'est pas toujours le cas, ainsi qu'il résulte des recherches du Prof. Gordan sur les nombres des covariants 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 equations 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 unsoughtfor 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. and thereby to advance the standards of the Science of Algebraical Forms to the most advanced point that has hitherto been reached. The stone that was rejected by the builders has become the chief corner-stone of the building. I shall for greater clearness begin with the case of a single binary quantic (a, b, c,..., lxx, y). Any rational integral function of the elements a, b, c,... which remains unchanged 7 in value when for them are substituted the elements of the new quantic obtained by putting a+hy instead of x in the original one, I call a Differentiant in a to the given quantic. By a differentiant of a given weight w and order j, I mean one in every term of which the combination of the elements is of the jth order and the sum of their weights w, the weights of the successive elements (a, b, c,...) themselves being reckoned as 0, 1, 2,... i respectively. The proposition to be proved is, that the number of arbitrary constants in the most general expression for such differentiant is the difference between the number of ways in which w can be made up with the integers 0, 1, 2, 3,...i (repetitions allowable), less the number of ways in which w-1 can be made up with the same integers. We may denote these two numbers by (wi, j), (w−1 : i,j) respectively, and their difference by A(w: i,j). Then, if we call the number of arbitrary constants in the differentiant of weight w and order j belonging to a binary quantic of the ith order D(w: i, j), the proposition to be established is that D(w: i,j)=▲(w: i,j). Let us use to denote the operator Then it is well known that the necessary and sufficient condition for D being a differentiant in x is that the identity QD=0 be satisfied. Let us study the relations of and O in respect to D. In the first place, let U be any rational integral function of the elements of order j and weight w; then I say that N.0.U-0.N.U=(ij—2w)U. For if we use to signify the act of pure differential operation, it is obvious that 2.0.U=(N×O)U+(Q*O)U, ..N.O.U—ON.U=((N*O)—(0*N))U d d d db dc db-2(i-1) c ac d d =ia +(i−2)b + (i−4)c —(i—2)k ak da -21 dk 'dľ If now par.ba.c"...l, where p is a number, be any term in . U, we have i. e. p+q+r+...+t=j } ... Ω.Ο.U-0.Ω.U, = (ij—2w)U, as was to be proved. If now for U we write D a differentiant in a, we have QD=0, and therefore 2.0(0.D)—0.(O.D)=(ÿj−2(w+1))0.D; for O.D is of the weight w+1; .. N2. 02. D=N.OSD+(8—2)0.0.D =(28-2).O.D =8(28-2)D. Similarly it will be seen that No3. O3.D=8(28—2)(38—6)D, and in general N2. O2.D=8(28—2)(38—6)... (qd—(q2+q))D =(1.2.3...q)(8.8—1 8—2...♪-q-1)D, the successive numbers 8, 28-2, 38-6, &c. being the successive sums of the arithmetical series 8, 8-2, 8-4, 8-6, &c. To find the most general differentiant in question, we must take every combination of the elements whose weight is w and order j, of which the number is obviously (w: i, j), and prefix an indeterminate constant to each such combination; then operating upon this form with 2, we shall reduce its weight by unity, and shall obtain as many combinations of this reduced weight (the order j remaining unchanged) as there are units in (w-1: i,j). Each of these combinations will have for its coefficient a linear function of the assumed indeterminate coefficients; and in order to satisfy the identity D=0, each such linear function must be made equal to zero. There are therefore (w: i,j) quantities connected by (w-1: i,j) homogeneous equations. Supposing the equations to be independent, the number of the indeterminate coefficients left arbitrary is obviously the difference between these quantities, viz. A(w: i, j). The difficulty consists in proving this independence-a difficulty so great that I think any one attempting to establish the theorem, as it were by direct assault, in this fashion, would find that he had another Plevna on his hands. But a position that cannot be taken by storm or by sap may be turned or starved into surrender; and this is how we shall take our Plevna. Be the equations of condition linearly independent or not, it is obvious that we must have D(w: i, j) equal to or greater than A(w: i, j). I shall show by aid of a construction drawn from the resources of the "Imaginative Reason," and founded on the reciprocal properties that have just been exhibited by the famous O and 2, that this latter supposition, of the first member of the equation being greater than the second, is inadmissible and must be rejected. Observe that (0: i,j), the number of ways of making up 0 with j combinations of 0, 1, 2, ..... ¿, is 1 ; also that D(0: i,j), the number of arbitrary constants in the most general differentiant in a to the quantic (a, b, c,... (x, y) of order j and weight 0 is also 1; for such differentiant is obviously λa". Thus we have for all values of w, and also D(w: i,j)= or>(w : i,j)—(w−1 : i, j), D(0: i,j)=(0: i,j); .. D(w: i, j)+D(w−1 : i, j)+D(w−2 : i, j) + ... +D(0: i,j)= or >(w: i, j). If in the above condition, for any assumed value of w, > is the sign to be employed, then the equation D(w: i, j)=▲(w: i,j) cannot be satisfied for all values of w. If, on the other hand, 2 2 > is not the sign to be employed, then this equation, for every value of w, commencing with the assumed one down to 0, must be satisfied. The greatest value of w for given values of i, j, it is well known, is 2 for ij even, and for ij odd. Let us give to w this maximum value in the above "greater or equal" relation; for brevity, denote the differentiants whose types are[w, i, j], [w-1,i,j] ... by [w], [w-1], [w—2], &c. respectively, i and j being regarded as constants. It will be convenient to substitute for the number of arbitrary constants in any of these differentiants the same number of linearly independent specific values of them; so that we shall have D(w: i,j) of linearly independent [w]'s, D(w-1: i,j) of linearly independent [w-1]'s, and so on. Now, instead of the D(w-qi, j) differentiants [w-q], let us substitute the same number of the derived forms (O'[w-q])'s. I shall prove that the quantities (all of the same weight w) thus obtained are linearly independent of one another. For (1) suppose that those belonging to any one set Oo. [w-q] are not independent, but are connected by a linear equation. Then, operating upon this equation with No. we shall obtain a linear equation between the quantities [w-q] for each quantity (Q. O. [w-g] being a numerical multiple of [w-q]), which is contrary to the hypothesis. Again, let there be a linear equation between the quantities contained in any number of sets of the form O'. [w-q] for which m is the greatest value of q. Then, operating upon this with ", it is clear that all the quantities in the sets for which q<m will introduce quantities of the form [w-q] where m-q>0, and which consequently vanish. There will be left, therefore, only quantities of the form [w-q], between which a linear equation would exist, contrary to hypothesis, as in the preceding case. Therefore all the quantities in all the sets are linearly independent. But these are all of the weight w, i.e. [or], and are therefore linear functions of the num 2 ber of ways in which the integers 0, 1, 2, 3,...i can be combined i and j together so as to give the weight w. Therefore being linearly independent, as just proved, their number cannot exceed this last-named number, i. e. cannot exceed (w: i,j). That is to say, D(w: i, j)+D(w−1 : i, j) + ... +D(0 : i, j) cannot exceed (w: i, j). Therefore every one of the equations |