Jan. 7, 1886J 



NA TURE 



So much — for time will not admit of more — concerning 

 pure reciprocants. 



Let me now say a few words en passant on Mixed 

 Reciprocants. 



Pure Reciprocants, we have seen, are the analogues of 

 Invariants, or else of the leading terms [for that is what 

 are Semi- or Subinvariants] of Covariantive expansions ; 

 each is subject to its own proper linear partial differential 

 equation. Mi.xed Reciprocants are the exact analogues 

 of the coefficients in such expansions other than those of 

 the leading terms. Starting from the leading terms as 

 the unit point, the coefficients of rank u are subject to a 

 partial differential equation of order u ; and just so, 

 mixed reciprocants, if involving r up to the power w, are 

 subject to a partial differential equation of that same 

 order. 



I have alluded to a peculiar class of mi.xed under the 

 name of " Orthogonal Reciprocants." They are distin- 

 guished, as I have proved, by the beautiful property that, 

 if differentiated with respect to r, the result must be itself 

 a Reciprocant. In Chart i you will see this illustrated in 

 the case of a mixed reciprocant (i -|- t'-)(J — jt^-, which 

 serves to indicate the existence of points of maximum 

 and minimum curvature. Its differential coefficient with 

 respect to t is the oft-alluded-to Schwarzian, transliterated 

 into the simpler notation. Proceeding in the inverse 

 order — of Integration instead of Differentiation — I call 

 your attention to a mixed reciprocant, of a very simple 

 character, one which presents itself at the very outset of 

 the theory, viz. — 



TC — lab, 



which, integrated in respect to t between proper limits, 

 yields the elegant orthogonal reciprocant — 



(7-- -\- \)c- loalir + i^a'K 



Expressed in the ordinary notation, this, equated to 

 zero, takes the form — 



(S 



+ 1 



t/'v 



dv d-v d^v /d'v\> 



' dv' dl-' ' tt^^^^^dP' 



Mr. Hammond has integrated thii, treated as an 

 ordinary differential equation, and has obtaineJ the 

 complete primitive expressed through the medium of two 

 related Hyper-Elliptic Functions connecting the variables 

 X and J' (see Chart 3). It may possibly turn out to be the 

 case that every mixed reciprocant is either itself an 

 Orthogonal Reciprocant, or by integration, in respect to r, 

 leads to one. 



It will of course be understood that, in interpreting 

 equations obtained by equating to zero an Orthogonal 

 Reciprocant, the variables must be regaided as represent- 

 ing not general but rectangular Cartesian coordinates. 



Here seems to me to be the proper place for pointing 

 out to what extent I have been anticipated by M. Halphen 

 in the discovery of this new world of Algebraical Forms. 

 When the subject first dawned upon my mind, about the 

 end of October or the beginning of November last, I was 

 not aware that it had been approached on any side by 

 any one before me, and believed that I was digging into 

 absolutely virgin soil. It was only when I received M. 

 Halphen's letter, dated November 25, in relation to the 

 Mongian business already referred to, accompanied by a 

 presentation of his memoirs on Differential Invariants, 

 that I became aware of there existing any link of con- 

 nection between his work and my own. A Differential 

 Invariant, in the sense in which the term is used by 

 M. Halphen, is not what at first blush I supposed it to 

 be, and as in my haste to repair what seemed to me an 

 omission to be without loss of time supplied, I wrote to M. 

 Hermite it was, in a letter which has been or is about to be 

 inserted in the Comptes rendits o( the Institute of France ; 

 it is not, I say, identical with what I have termed a 

 general pure reciprocant, but only with that peculiar species 



of Pure Reciprocants to which I have in a preceding 

 part of this lecture referred as corresponding and point- 

 ing to Projective Singularities. In his splendid labours in 

 this field Halphen has had no occasion to construct or 

 concern himself with that new universe of forms viewed 

 as a whole, whether of Pure or Mixed Reciprocants,- 

 which it has been the avowed and principal object of this 

 lecture to bring under your notice. 



I anticipate deriving much valuable assistance in the 

 vast explorations remaining to be made in my own 

 subject from the new and luminous views of M. Halphen, 

 and possibly he may derive some advantage in his turn 

 from the larger outlook brought within the field of vision 

 by my allied investigations. 



Let me return for a moment to that simplest class of 

 pure reciprocants which I hive called protomorphs. Each 

 of these will be found (as may be shown either by a direct 

 process of elimination, or by integrating the equations 

 obtained by equating them severally to zero, regarded as 

 ordinary differential equations between .1" and f) each of 

 these, I say, will be found to represent some simple kind 

 of singularity at the point (.f, J') of the curve to which 

 these co-ordinates are supposed to refer. Thus, for 

 instance, No. i marks a single point of inflexion ; No. 2, 

 points of closest contact with a common parabola ; No. 3, 

 what our Cayley has called sextactic points, referring to a 

 general conic ; No. 4, points of closest contact with a 

 common cubical parabola ; and so on. The first and 

 third, it will be noticed, represent projective singularities, 

 and as such, in M. Halphen's language, would take the 

 name of Differential Invariants. The second and fourth, 

 having reference to the line at infinity in the plane of the 

 curve, are of a non-projective character, and as such 

 would not appear in M. Halphen's system of Differential 

 Invariants. It is an interesting fact that every simple 

 parabola, meaning one whose equation can be brought 



under the form/ = .x^' , corresponds to a linear function 

 of a square of the third, and the cube of the second proto- 

 morph, and consequently will in general be of the sixth 

 degree. In the particular case of the cubical parabola, 

 the numerical parameter of this equation is such that the 

 highest powers of i cancel each other so that the form 

 sinks one degree, and becomes represented by the Quasi- 

 Discriininant, No. 4. 



This simple instance will serve to illustrate the intimate 

 connection which exists between the projective and non- 

 projective reciprocants, and the advantage, not to say 

 necessity, of regarding them as parts of one organic 

 whole. 



It would take me too far to do more than make the 

 most cursory allusion to an extension of this theory 

 similar to that which happens when in the ordinary 

 theory of invariants we pass from the consideration of a 

 single Ouantic to that of two or more. There is no diffi- 

 culty in~finding the partial differential equation to double 

 reciprocants which, as far as I have as yet pursued the 

 investigation, appear to be functions of a, b, c, . . . ; 

 a', b', c', . . ; and of (r - t'). 



The theory of double reciprocants will then include as 

 a particular case the question of determining the singu- 

 larities of paired points of two curves at which their 

 tangents are parallel, and consequently the theory of 

 common tangents to two curves and of bi-tangentj to a 

 single one. 



I think I may venture to say that a general pure 

 multiple reciprocant which marks off relative singularities, 

 whether projective or non-projective, of a group of curves, 

 is a function of the second and higher differential 

 derivatives appertaining to the several curves of the 

 group, and of the differences of the first derivatives, 

 whereas in a mi.xed multiple reciprocant these last- 

 named differences are replaced by the first derivatives 

 themselves. As a particular case, when the group 



