224 



NA TURE 



{yan. 7, 1886 



opposite Peaucellier cells, coupled toe-and-toe together, 

 swing into motion, which would have been impossible 

 had not the two connected moving points each described 

 an accurate straight line, " the house rose at you." [The 

 lecture merely illustrated experimentally two or three 

 simple propositions of Euclid, Book III.]. 



The matter that I have to bring before your notice this 

 afternoon is one far bigger and greater, and of infinitely 

 more importance to the progress of mathematical science, 

 than any of those to which I have just referred. No subject 

 during the last thirty years has more occupied the minds 

 of mathematicians, or lent itself to a greater variety of 

 applications, than the great theory of Invariants. The 

 theory I am about to expound, or whose birth I am about 

 to announce, stands to this in the relation not of a younger 

 sister, but of a brother, who, though of later birth, on the 

 principle that the masculine is more worthy than the 

 feminine, or at all events, according to the regulations of 

 the Salic law, is entitled to take precedence over his elder 

 sister, and e.xercise supreme sway over their united realms. 

 Metaphor apart, I do not hesitate to say that this theory, 

 minor iiatu potcstate major, infinitely transcends in the 

 e.\tent of its subject-matter, and in the range of its appli- 

 cations, the allied theory to which it stands in so close a 

 relation. The very same letters of the alphabet which 

 may be employed in the two theories, in the one may be 

 compared to the dried seeds in a botanical cabinet, in the 

 other to buds on the living branch ready to burst out into 

 blossom, flower, and fruit, and in their turn supply fresh 

 seed for the maintenance of a continually self-perpetuating 

 cycle of living forms In order that I may not be con- 

 sidered to have lost myself in the clouds in making such 

 a statement, let me so far anticipate what I shall have to 

 say on the meaning of Reciprocants and their relation 

 to the ordinary Invariantive or Covariantive forms by 

 taking an instance which happens to be common (or at 

 least, by a slight geometrical adjustment, may be made 

 so) to the two theories. I ask you to compare the form 



d-d — 2iil>c -\- 2//' 



as it is read in the light of the one and in that of the 

 other. In the one case the «, d, c, d stand for the co- 

 efficients of a so-called Binary Ouantic, and its evan- 

 escence serves to express some particular relation be- 

 tween three points lying in a right line. In the other 

 case the letters are interpreted to mean the successive 

 differential derivatives of the 2nd, 3rd, 4th, sth orders of 

 one Cartesian co-ordinate of a curve in respect to the other. 

 The equation expressing this evanescence is capable 

 of being integrated, and this integral will serve to denote 

 a relation between the two co-ordinates which furnishes 

 the necessary and sufficient condition in order that the 

 point of the curve of any or no specified order (for it may 

 be transcendental) to which the co-ordinates may refer, 

 may admit of having, at the point where the condition is 

 satisfied, a contact with a conic of a higher order than the 

 common. In the one case the letters employed are dead 

 and inert atoms ; in the other they are germs instinct 

 with motion, life, and energy. 



A curious history is attached to the form which I have 

 just cited, one of the simplest in the theory, of which the 

 narrative may not be without interest to many of my 

 hearers, even to those whose mathematical ambition is 

 limited to taking a high place in the schools. 



At pp. 19 and 20 of Boole's " Differential Equations'' 

 (edition of 1859) the author cites this form as the lefthand 

 side of an equation which he calls the " Differential 

 Equation of lines of the second order," and attributes it 

 to Monge, addmg the words, " But here our powers of 

 geometrical interpretation fail, and results such as this 

 can scarcely be otherwise useful than as a registry of 

 integrable forms." In this vaticination, which was quite un- 

 called for, the eminent author, now unfortunately deceased, 

 proved himself a filse prophet, for the form referred to 



is among the first that attracts notice in crossing the 

 threshold of the subject of Reciprocants, and is but one 

 of a crowd of similar and much more complicated ex- 

 pressions, no less than it, susceptible of geometrical 

 interpretation and of taking their place on the register of 

 integrable forms. A friend, with whom I was in commu- 

 nication on the subject, and whom I see by my side, 

 remarked to me, in reference to this passage ; — " I cannot 

 help comparing a certain passage in Boole to Ezekiel's 

 valley of the clry bones : ' The valley was full of bones, 

 and lo, they were very dry.' The answer to the question, 

 ' Can these bones live ? ' is supplied by the advent of the 

 glorious idea of the Reciprocants ; and the grand invo- 

 cation, ' Come from the four winds, O breath, and breathe 

 upon these slain, that they may live,' may well be used 

 here. That they will ' live and stand up upon their feet an 

 exceeding great army' is what we may expect to happen." 

 This, as you will presently see, is just what actually has 

 happened. 



Not knowing where to look in Monge for the implied 

 reference, I wrote to an eminent geometer in Paris to 

 give me the desired information ; he replied that the 

 thing could not be in Monge, for that M. Halphen, who 

 had written more than one memoir on the subject of the 

 differential equation of a conic, had made nowhere any 

 allusion to Monge in connection with the subject. Here- 

 upon, as I felt sure that a reference contained in repeated 

 editions of a book in such general use as Boole's " Difler- 

 entialEquations"was not likely to be erroneous, I addressed 

 myself to M. Halphen himself, and received from him a 

 reply, from which I will read an extract : — 



" En premier lieu, c'est une chose nouvelle poui moi 

 que r^quation differenlielle des coniques se trouve dans 

 Boole, dont je ne connais pas I'ouvrage. Je vais, bien 

 cntendu, le consulter avec curiosite. Ce fait a echappd a 

 tout le monde ici, et Ton a cru generalement que j'avais 

 le premier donn^ cette Equation. A'/'/ sub sole novi ! II 

 m'est naturellement impossible de vous dire oii la meme 

 equation est enfouie parmi les ceuvres de Monge. Pour 

 moi, c'est dans Le Journal dc Matli. (1S76), p. 375, que 

 j'ai eu, je crois, la premiere occasion de developper cette 

 Equation sous la forme meme que vous citez ; et c'est 

 quand je I'ai employde, I'annee suivante, pour le probleme 

 sur les lots de Kepler {Com pies rcndus, 1877, t. Ixxxiv. 

 P- 939)1 que M. Bertrand I'a remarqu^e comme neuve. 

 Ce qui vous int^resse plus, c'est de connaitre la fonne 

 simplifii^e sous laquelle j'ai donnd plus tard cette (Equation 

 dans le Bulletin de la Societd Math(5matique. C'est sous- 

 cette derniere forme que M. Jordan la donne dans son 

 cours de I'Ecole Polytechnique " (t. i. p. 53). 



All my researches to obtain the passage in Monge 

 referred to by Boole have been in vain.' 



I will now proceed to endeavour to make clear to you, 

 what a Reciprocint means : the above form, which may 

 be called the Alongian, would afford an example by which 

 to illustrate the term ; but I think it desirable to begin 

 with a much easier one. Consider then the simple case 

 of a single term, the second derivative of one variable, y, 

 in respect to another, .v. Every tyro in algebraical geo- 

 metry knows that this, or rather the fact of its evanescence, 

 serves to characterise one or more points in a curve u-hich 

 possess, so to say, a certain indelible and intrinsic cha- 

 racter, or what is technically called a singularity ; in this 

 case an inflexion such as exists in a capital S, or Hogarth's 

 line-of beaut)-. 



If we invert the two variables, exchanging, that is to say, 

 one with the other, the fact of this indelibility draws with 



I Search h.-is been maie in the collected works of Monge and in manuscripts 

 of his own or Prony in the library of the Institute, but without effect. I 

 liave also made application to tlie Universal Information Society, who 

 ' every conceivable question," but nothing has so far 

 ntil the citation from Monge is verified it will be safer 

 so-called Mongian as the Eoole-Mongian. It may 

 ting-point of the Differential Invariant 'theory", as the 

 detper-lyins and more comprehensive Reciprocant 



undertake to 



