Jan. 7, i«86j 



NA TURE 



229 



inequality. If this conjecture is verified it is not merely 



(y- - 2) 



IV, which is never negative : 



and when this is zero, the form may be said to be com- 

 plete.i There would then be thus complete forms in each 

 of the two theories ; in the earlier one they take a special 

 name : this is the only difference. 



We have spoken of Pure Reciprocants as being 

 either projective or non-projective, but so far have 

 abstained from particularising the external characters 

 by which the former may be distinguished from the 

 latter. I have good reason to suspect that the former are 

 distinguished from the latter by being Binariants ; that, 

 in addition to being subject to annihilation by the 

 operator V, they are also subject to annihilation by the 

 Vermicular operator when made special by the use of the 

 numerical multipliers 3, S, 15 . . . above alluded to, or in 

 other words (as previously mentioned incidentally) are 

 subject to satisfy two simultaneous partial differential 

 equations instead of only one.- Projective Reciprocants 

 we have seen are disguised or masked Ternary Covariants 

 — Covariants in the grub, the first undeveloped state. Now 

 ternary covariants are capable, it may or may not be 

 generally known, of satisfying 6 reducible to 2 simul- 

 taneous Partial Differential Equations, and at first sight it 

 might be surmised that nothing would be gained by the 

 substitution of the two new for the two old simultaneous 

 partial differential equations. But the fact is not so. for 

 the old partial differential equations are perfectly un- 

 manageable, or at least have never, as far as I know, been 

 handled by any one, for they have to do with a triangular 

 heap, whereas the new ones are solely concerned with a 

 linear series of co-efficients. 



I have alluded to there being a particular form common 

 to the two theories. In the one theory it is the Mongian 

 alluded to in the correspondence, which has been 

 read, with M. Halphen. In the other it is the source 

 of the skew covariant to the cubic. If the latter be subjected 

 to a sort of MacMahonic numerical adjustment, it becomes 

 absolutely identical with the former. Let us imagine that 

 before the invention of Reciprocants an Algebraist hap- 

 pened to have had both forms present to his mind, and had 

 thought of some contrivance for lowering the coefficients 

 of the Mongian written out with the larger coefficients, 



t If this should turn out to be tnie, the "crude generating fraction " for 

 reciprocants would be almost identical with that of in- and co-variants of the 

 same extenty. The denominators would be absolutely identical ; as regards 

 the numerators, while that for invariantive forms is i — a'^x'~ the numerator 

 for reciprocants would be i — a'-.x'^. As I write abroad and from memory 

 there is just a chance that the index of rt here given may be erroneous. 



- As already stated in a previous footnote this conjecture is fully confirmed, 

 my own proof having been corroborated (if it needed corroboration) by another 

 entirely different one invented by M. Halphen, who fully shares my own 

 astonishment at the fact of there being forms (half-horse, half alligator) at 

 once reciprocants and sub-invar ants, and as such satisfying two simultaneous 

 partial differential equations. 



If instead of denoting the successive differential derivatives (starting from 



the second a,b,c,... we call them —, > , ... the two An- 



1.2 1.2.3 1.2.3.4' 

 nihilators will be 



all, + 2bSc + 3<-5rf -I- 4</S, -I- ... and 



^"-St, + 5(7fSt -1- 6{ad + &)5rf -I- ^(ae + lid + i j5, -f . . . 



the latter being my new operator, the Reciprocator /-' accommodated to the 

 above-stated change of notation for the successive differential derivatives. 

 Hardly necessary is it for me to point out in explanation of the semi-sums 



, . . . that we may write the MacMahonised V under the form 



,^a-l^, + t,{ac + ca)Sc+(>(a<: + l>c)Sd+^(ac + h^+c^ + M + ca)5t+ . . . 

 It is to be presumed that in addition to mixed reciprocants (the ocean into 

 which flows the .sea of pure reciprocants, as into that again empties itself the 

 river of projective reciprocants) there may exist a theory of forms in which j' as 



well as -y- will appear, or, so to say, doubly mixed reciprocants, the most 



general of all, in which case we must speak of the content of these as the 

 ocean and of the others as sea, river, and br. ok. Curious is it to reflect that 

 in the theory which as it exists comprises Invariantives, Reciprocants, and 

 Invariantive Reciprocants or Reciprocant Invariantives, the order of dis- 

 covery was (i) Invariantives (Eisenstein, Boole, &c.) ; (2) Invariantive Reci- 

 procants (Monge and Halphen); (3) Reciprocants (Schwarz, the authcrof 

 this lecture). 



and had thus stumbled upon this striking fact. It could 

 not have failed to vehemently arouse his curiosity, and he 

 would have set to work to discover, if possible, the cause 

 of this coincidence. He would in all probability have 

 addressed himself to the form which precedes the source 

 alluded to in the natural order of genesis, and have 

 applied a similar adjustment to the much simpler form, 

 ac - l>- : having done so he would have tried to discover 

 to what singularity it pointed — but his efforts to do so we 

 know must have been fruitless, and he would have felt dis- 

 posed to throw down his work in despair, for the inter- 

 mediate ideas necessary to make out the parallelism would 

 not have been present to his mind. So long as we confine 

 ourselves to Differential Invariants, i.e. to projective pure 

 reciprocants, we are like men walking on those elevated 

 ridges, those more than Alpine summits, such as I am told ' 

 exist in Thibet, where it may be the labour of days for two 

 men who can see and speak to each other to come together. 

 Reciprocants supply the bridge to span the yawning ravine 

 and to bring allied forms into direct proximity. 



I have spoken of mixed reciprocants as being subject to 

 satisfy not a linear partial differential equation, but one of 

 a higher order dependent on the intensity, so to say, of its 

 mixedness — the highest power, that is to say, of the first 

 diffci-ential derivative which it contains, and it might there- 

 fore be supposed that these forms are much more difficult 

 to be obtained than pure reciprocants. But the fact is 

 just the reverse, for as I discovered in the very infancy of 

 the inquiry, and have put on record in the September or 

 October number of the Mathematical Messenger, mixed 

 reciprocants may be evolved in unlimited profusion by the 

 application of simple and explicit processes of multiplica- 

 tion and difterentiation. From any reciprocant whatever, 

 be it mixed or pure, new mixed ones may be educed 

 infinitely infinite in number, inasmuch as at each stage 

 of the process, arbitrary functions of the first differential 

 derivative may be introduced. 



The wonderful fertility of this method of generation 

 excited warm interest on the part of one of the greatest 

 of living mathematicians, the e.xpression of which acted 

 as a powerful incentive to me to continue the inquiry. 

 They may be compared with the shower of December 

 meteors shooting out in all directions and covering the 

 heavens with their brilliant trains, all diverging from one 

 or more fixed radiant-points, the radiant-point in the 

 theory before us being the particular form selected to be 

 operated upon. 



The new doctrine which I have endeavoured thus im- 

 perfectly to adumbrate has taken its local rise in this 

 University, where it has already attracted some votaries to 

 its side, and will, I hope, eventually obtain the co-opera- 

 tion of many more. I have ventured with this view to 

 announce it as the subject of a course of lectures during 

 the ensuing term. 



When I lately had the pleasure of attending the new 

 Slade Professor's inaugural discourse, I heard him promise 

 to make his pupils participators in his work by painting 

 pictures in the presence of his class. I aspire to do more 

 than this — not only to paint before the members of 

 my class, but to induce them to take the palette and 

 brush and contribute with their own hands to the 

 work to be done upon the canvas. Such was the plan I 

 followed at the Johns Hopkins University, during my 

 connection with which I may have published scores of 

 Mathematical articles and memoirs in the journals of 

 America, England, France, and Germany, of which pro- 

 bably there was scarcely one which did not originate 

 in the business of the class-room ; in the composition of 

 many or most of them I derived inestimable advant.agc 

 from the suggestions or contributions of my auditors. Il 

 was frequently a chase, in which I started the fox, in which 



I I thinkmyinformant was my friend Dr. Inglis, of the AthenEeum Club, who 

 some time ago undertook a journey in the Himalayas in the hopes of coming 

 upon the traces of a lost religion which he thought he had reason to believ., 

 existed among mankind in the pre-OIacial period of the earth's history. 



