2 26 



NA TURE 



\yoin. 7, iSS6 



jective or non-projective, are indicated by pure recipro- 

 cants, and are subject to the same Partial Differential 

 Equation ; just as, in the Theory of Binary Ouantics, 

 Invariants, although under one aspect they may be re- 

 garded as a self-contained special class, admit of being 

 and are most advantageously studied in connection with, 

 and as forming a part of, the whole family of forms com- 

 monly known by the name of " semi-, or subinvariants," 

 but which I find it conduce to much greater clearness of 

 expression and avoidance of ambiguity or periphrasis to 

 designate as Binariants. 



The question may here be asked, How, then, are 

 projective and non-projective pure reciprocants to be 

 discriminated by their external characters ? 



1 believe that I know the answer to this question, 

 which is, that the former arc subject to satisfy a second 

 partial differential equation of a certain simple and fami- 

 liar type, but this is a matter upon which it is not neces- 

 sary for me to enter on the present occasion.' It is 

 enough for our present purpose to remark that every 

 projective pure reciprocant must, so to say, be in 

 essence a masked ternary covariant. For instance, if we 



take the simplest of all such, viz. a, i.e. ~^, we have 



dx- 



d'-y 

 d.v^ 



■^m- 



which, for facility of reference, let me call M. Obviously 

 we might instead of ^i = o substitute M = o to mark an 

 inflexion. And now if we write * as the completed form 

 of (^, when made homogeneous by the substitution of .~ 

 for unity ; and if we suppose it to be of ii dimensions in 

 x,y, z, and call its Hessian H, we shall obtain the syzygy 



(„ - I)-(^J . -+^+[J^. ■ ^. - [-^,) l*=°- 



Hence the system * = o, « = o, will be in effect the same 

 as the system * = o, // = o, and in this sense a may be 

 said to carry // as it were in its bosom. And so in 

 general every pure projective reciprocant may, in the 

 language of insect transformation, be regarded as passing, 

 so to say, first from the grub to the pupa or chrysalis, and 

 from this again, divested of all superfluous integuments, 

 to the butterfly or imago state. 



Non-projective pure reciprocants undergo only one such 

 change. There is no possibility of their ever emerging 

 into the imago— their development being finally arrestetl 

 at the chrysalis stage. 



It would, I think, be an interesting and instructive task 

 to obtain the imago or Hessianised transformation of tlic 

 Mongiin, but I am not aware that any one has yet done, 

 or thought of doing, this.- It seems to me that by substi- 

 tuting Reciprocants in lieu of Ternary Covariants we are 

 as it were stealing a dimension from space, inasmuch as 

 Reciprocants, /.c. Ternnry Covariants in their undeveloped 

 state,are closely allied to, and march pan' passu with, the 

 familiar forms which appertain to merely binary quantics. 



1 will now proceed to bring before your notice the 

 general partial differential equation which supplies the 



* In Paris, from which 1 correct ths proofs, I have succeeded in reducing 

 this c mjeclure 10 a certainty and in estabhshing the marvellous fact that 

 every Projective Reciprocant, or, which is the same thing, every Differential 

 Invariant, is, at the same time, an Ordinarj' Subinvariant. U hus a differ- 

 ential invariant (or projective reciprocant) may be regarded as n sin°ic per- 

 sona-litychthed it'ith t^^vo distinct natures— \\\^\. of a reciprocant and that 

 of a suljinvariant. 



° M. Halphen informs me that this has been done by Cayley in the PHI 

 Trans, for 1865, and subsequently in a somewhat simplified form by Painvtn, 

 Compies rendus, 1874. But neitller of these authors s'^ems to have had the 

 Boole- Mongian objectively before them, so that a slight supplemental com- 

 putation is wanting to establish the equation between it and the function 

 which either of them finds to vanish at a sextactic point 



necessary and sufficient conditio;! to which all pure 

 reciprocants are subject. 



It is highly convenient to' denote the successive 

 derivatives 



d y d y d y 

 dx-' Zv^' dl-*' ' ' ' 

 by the siinple letters a,b,c, . . . 



The first derivative ~- plays so peculiar a part in this 

 dx 

 theory that it is necessary to denote it by a letter standing 

 aloof from the rest, and I call it t. This last letter, I need 

 not say, does not ma':e its appearance in any pure 

 reciprocant. This being premised, I invite your atten- 

 tion to the equation in question, in which you will perceive 

 the symbols of operation are separated from the object to 

 be operated upon. 



Writing V = ia"-h -\- \oahh, + {i^ac -f- lOi^-)S., + . . . . 

 and calling any pure reciprocant R, 



VR = o 



is the equation referred to. 



I cannot undertake, within the brief limits of time allotted 

 to this lecture, to explain how this operation, or, as it 

 may be termed, this annihilator f'is arrived at. The table 

 of binomial coefficients, or rather half series of binomial 

 coefficients, shown in Chart 4, will enable you to see what 

 is the law of the numerical coefficients of its several terms. 

 Let the words Wi'ight, degree, extent (extent, you will 

 remember, means the number of places by which the most 

 remote letter in the form is separated from the first letter 

 in the alphabet) of a pure reciprocant signify the same 

 things as they would do if the letters a, b, e, . . . referred, 

 according to the ordinary notation, to Binariants instead 

 of to Reciprocants. The number of binariants linearly 

 independent of each other whose weight, extent, and 

 order are w, i, j is given by the partition formula 

 {w ; I, J) — {it' — I ; t\j) where in general (71:/; /,y) means 

 the number of ways of partitioning 7C> into / or fewer parts 

 none greater than j. It follows immediately from th6 

 mere form of Kthat the corresponding formula in the case 

 of Reciprocants of a given type ic.t.j will be {w : i,j) — 

 {w — I \t-\-\,j) the augmentation of / in the second 

 term of the formula being due to the fact that, whereas in 

 the partial differential equation for Binariants it is the 

 letters themselves which appear as coefficients, it is 

 quadratic functions of these in the case of Reciprocants. 

 From the form of f-^ we may also deduce a rigorous de- 

 monstration of the existence of Reciprocants strictly 

 analogous to those with which you are familiar in the 

 Binarian Theory, v/hich are pictured in Chart 2, and are now 

 usually designated as Protomorphs, as being the forms by 

 the interweaving of which with one another (or rather by 

 a sort of combined process of mixture and precipitation), 

 all others, even the irreducible ones, are capable of being 

 produced. The corresponding forms for Reciprocants 

 you will see exhibited in the same table. Each series of 

 Protomorphs may of course be indefinitely extended as 

 more and more letters arc introduced. In the table I 

 have not thought it necessary to go beyond the letter § 

 You also know that besides Protomorphs there are other 

 irreducible forms, the organic radicals, so to say, into 

 which e\'ery compound form may be resoh-ed, always 

 limited in number, whatever the number of letters or 

 primal elements we may be dealing with. The same 

 thing happens to Reciprocants as you will notice in the 

 comparative table in Chart 2. Without going into particu- 

 lars, I will ask you to take from me upon faith the assur- 

 ance that there is no single feature in the old familiar 

 theory, whether it relates to Protomorphs, to Ground- 

 forms, to Perpetuants, to Factorial constitution, to Gene- 

 rating Functions, or whatever else sets its stamp upon 

 the one, which is not counterfeited by and reproduced in 

 the parallel theory. 



