228 



NATURE 



\yan. 7, iJ 



dwindles to an individual and there is only one t, this 

 letter disappears altogether from the form, for there are 

 no differences of a single quantity. 



In the chart (marked No. 2) you will see the table of 

 Protomorphs carried on as far as the letter g inclusive, 

 and will not fail to notice what may be termed the higher 

 organisation of Reciprocantive as compared with ordinary 

 Invariantive Protomorphs ; the degrees of the latter 

 oscillate or librate between the numbers 2 and 3, whereas 

 in the former the degree is variable according to a certain 

 transcendental law dependent on the solution of a problem 

 in the Partition of Numbers. Another interesting differ- 

 ence between general Invariants and general Pure 

 Reciprocants consists in the fact that, whilst the number 

 of the former ultimately (i.e. when the extent is inde- 

 finitely increased) becomes indefinitely great, that of the 

 latter is determinate for any given degree even for an 

 infinite number of letters. 



In carrying on the table of protomorphs up to the letter 

 h (see Chart 6) a new phenomenon presents itself, to which, 

 however, there is a perfect parallel in the allied theory. 

 An arbitrary constant enters into the form, its general 

 value being a linear function of £/and TK (for which see 

 Chart 6). But this is not all. If you examine the terms in 

 both U and W (there are in all twelve such) you will 

 find that these twelve do not comprise all of the same type 

 to which they belong. There is a Thirteenth (a banished 

 Judas), equally a priori entitled to admission to the group, 

 but which does not make its appearance among them, 

 viz. b^d. I rather believe that a similar phenomenon of 

 one or more terms, whose presence might be expected, but 

 which do not appear, presents itself in the allied invarian- 

 tive theory, but cannot speak with certainty as to this 

 point, as the circumstance has not received, and possibly 

 does not merit, any very particular attention. 



Still, in the case before us, this unexpected absence of a 

 member of the family, whose appearance might have been 

 looked for, made an impression on my mind, and even 

 went to the extent of acting on my emotions. I began 

 to think of it as a sort of lost Pleiad in an Algebraical 

 Constellation, and in the end, brooding over the subject, 

 my feelings found vent, or sought relief, in a rhymed 

 effusion, a jeii de sottise, which, not without some appre- 

 hension of appearing singular or extravagant, I will venture 

 to rehearse. It will at least serve as an interlude, and 

 give some relief to the strain upon your attention before I 

 proceed to make my final remarks on the general theory. 



TO A MISSING MEMBER 

 Of a Family Group of Terms in an Alj;d>i-aical ForinuUi 



Lone and discarded one ! divorced by fate, 



Far from thy wished-for fellows — whither art flown ? 



Where lingerest thou in thy bereaved estate, 



Like some lost star, or buried meteor stone ? 



Thou mindst me much of that presumptuous one 



Who loth, aught less than greatest, to be great, 



From Heaven's immensity fell headlong down 



To live forlorn, self-centred, desolate : 



Or who, new Heraklid, hard exile bore, 



Now buoyed by hope, now stretched on rack of fear. 



Till throned Aslrsea, wafting to his ear 



Words of dim poitent through the .Atlantic roar. 



Bade him " the sanctuary of the Muse revere 



And strew with flame the dust of Isis' shore." 



Having now refreshed ourselves and bathed the tips of 

 our fingers in the Pierian spring, let us turn back for a few 

 brief moments to a light banquet of the reason, and enter- 

 tain ourselves as a sort of after-course with some general 

 reflections arising naturally out of the previous m.itter of 

 my discourse. It seems to me that the discovery of reci- 

 procants must awaken a feeling of surprise akin to that 

 which was felt when the galvanic current astonished the 

 world previously accustomed only to the phenomena of 

 machine or frictional electricity. The new theory is a 



ganglionic one : it stands in immediate and central relation 

 to almost every branch of pure mathematics — to Invariants, 

 to Differential Equations, ordinary and partial, to Elliptic 

 and Transcendental Functions, to Partitions of Numbers, 

 to the Calculus of Variations, and above all to Geometry 

 (alike of figures and of complexes), upon whose inmost re- 

 cesses it throws a new and wholly unexpected light. The 

 geometrical singularities which the present portion of the 

 theory professes to discuss are in fact the distinguishing 

 features of curves ; their teclmical name, if applied to the 

 human countenance, would lead us to call a man's eyes, 

 ears, nose, lips, and chin his singularities ; but these singu- 

 larities make up the character and expression, and serve 

 to distinguish one individual from another. And so it is 

 with the so-called singularities of curves. 



Comparing the system of ground-forms which it sup- 

 plies with those of the allied theory, it seems to me clear 

 that some common method, some yet undiscovered, deep- 

 lying, Algebraical principle remains to be discovered, 

 which shall in each case alike serve to demonstrate the 

 finite number of these forms (these organic radicals) for 

 any specified number of letters. The road to it, I believe, 

 lies in the Algebraical Deduction of ground-forms from 

 the Protomorphs.' Gordan's method of demonstration, so 

 difficult and so complicated, requiring the devotion of a 

 whole University semester to master, is inapplicable to 

 reciprocants, which, as far as we can at present see, do not 

 lend themselves to symbolic treatment. 



How greatly must we feel indebted to our Cayley, who 

 while he was, to say at least, the joint founder of the 

 symbolic method, set the first, and out of England little 

 if at all followed, example of using as an engine that 

 mightiest instrument of research ever yet invented by the 

 mind of man — a Partial Differential Equation, to define 

 and generate invariantive forms. 



With the growth of our knowledge, and higher views now 

 taken of invariantive forms, the old nomenclature has not 

 altogether kept pace, and is in one or two points in need of 

 a reform not difficult to indicate. I think that we ought 

 to give a general name — I propose that of Binariants — to 

 every rational integral form which is nullified by the 

 general operator 



\ad., + ,./;,/. + vcd^ + . . . . 



where \, }i,v, . . . are arbitrary numbers. 



This operator, I think, having regard to the way in 

 which its segments link on to one another, may be called 

 the Vermicular. 



Binariants corresponding to unit values of X, ^, i/, . . . 

 may be termed standard binariants. Those for which 

 these numbers are the terms of the natural arithmetical 

 series I, 2, 3, . . . Invariantive binariants, which may be 

 either complete or incomplete invariants ; these latter are 

 what are usually termed semi- or sub invariants. I may 

 presently have to speak of a third class of binariants for 

 which the arbitrary multipliers are the numbers 3, 8, 15, 

 24 , . . (the squares of the natural numbers each diminished 

 by unity) which, if the theorem I have in view is supported 

 by the event, will have to be termed Reciprocantive Bin- 

 ariants. But first let me call attention to what seems a 

 breach of the asserted parallelism between the Invarian- 

 tive and the Reciprocantive theories. In the former we 

 have complete and incomplete invariants, but we have 

 drawn no such distinction between one set of pure reci- 

 procants and another. A parallel distinction does however 

 exist. 



If we use ■ic, /, j to signify the weight, extent, and de- 

 gree of an invariantive form, iu is never less than the half 

 product of /y; when equal to it the form is complete. In 

 the case of reciprocants certain observed facts seem to 

 indicate that there exists an analogous but less simple 



1 See the section on the Algebraical Deduction of the Ground-forms of the 

 Qnintic in my memoir on SabTnvariants in the Amcricnn Journal o/Mixthc- 



