Jan. 7, 1 88 6 J 



NA TURE 



225 



it the consequence that in general these two reciprocal 

 functions must vanish together, and as a fact each is the 

 same as the other multiplied or divided by the third 

 power of the first derivative of the one variable with 

 respect to the other taken negatively. In this case we 

 are dealing with a single derivative and its reciprocal, 

 'ilie question immediately presents itself whether there 

 may not be a combination of derivatives possessing a 

 similar property. We know that no single derivative 

 except the second does. 



Such a combination actually presents itself in a form 

 which occurs in the solution of Differential Equations of 

 the second order, the form 



(iy ^ iP)' _ 3 /"/i'V-^ 



dx ' dx^ 2 \dy^' ' 

 which, fifter the name of its discoverer, Schwarz, we may 

 agree to call a Schwarzian (Cayley's " Schwarzian Deriva- 

 tiVe"i). If in this expression the .i-andj be interchanged, 

 its value, barring a factor consisting of a power of the 

 first derivative, remains unaltered, or, to speak more 

 strictly, merely undergoes a change of algebraical sign. 

 We may now arrive at the generalised conception of an 

 algebraical function of the derivatives of one variable in 

 respect to another, which, if we agree to pay no regard to 

 the algebraical sign, or to any power of the first derivative 

 that may appear as a factor, will remain unaltered when 

 the dependent and independent variables are interchanged 

 one with another ; and we may agree to call any such 

 function a Reciprocant. 



But here an important distinction arises — there are 



Reciprocants such as the one I first mentioned, -J^, or 



d.v- 

 such as the Mongian to which allusion has been made in 

 the letter from M. Halphen, in which the second and 

 higher differential derivatives alone appear, the first 

 differential derivative not figuring in the e.xpression. 

 These may be termed Pure Reciprocants. Thus I repeat 



dx--'' 



and 



^\d.v-/ dx^ ^^dx->- dx^ dx^^^ \dvV 



are pure reciprocants. Those from which the first 



derivative — is not excluded may be called Mixed Re- 



dx 

 ciprocants. An example of such kind of Reciprocants is 

 afforded by the Schwarzian above referred to. This dis- 

 tinction is one of great moment, for a little attention will 

 serve to make it clear that every pure reciprocant ex- 

 pressed in terms of -i- andj' marks an intrinsic feature or 

 singularity in the curve, whatever its nature may be, of 

 which X and_>' are the co-ordinates ; for if in place of the 

 variables (x,y) any two linear functions of these variables 

 be substituted, a pure reciprocant, by virtue of its recipro- 

 cantive character, must remain unaltered save as to the 

 immaterial fact of its acquiring a factor containing 

 merely the constants of substitution.- 



■ More strictly speaking tliis is Cayley's Schwarzian derivative cleared of 

 fractions— it may well be called the Schwarzian (see my note on it in the 

 ATat/tematical Messenger for Septemlier or October past). Prof. Greenhill 

 in regard to the Schwarzian derivative proper writes me as follows : — 



" I found the reference in a footnote to p. 74 of Klein's ' Vorlesungen Uber 

 das Ikosaeder, &c.,' in which Klein thanks Schwarz for sending him the 

 reference to a paper by Lagrange, ' Sur la construction des cartes geogra- 

 phiques' in the Nouveaitx Memoires de V Acadimie de Berlin^ 1779. Com- 

 pare also Schwarz's paper in Bd- 75 of Borchardt's Journal, where further 

 literary notices are collected together. Klein says further that in the 

 ' Sachsischen Gesellschaft von Januar 18S3,' he has considered the inner mean- 

 ing {innere I'edeittiing) of the differential equation (~t) ~^('')' 



.dn .. 



rt'here 1 



' d~ 



^^ There are two papers by Lagrange, one immediately following the other, 

 "Sur la construction des cartes geographiques," but I have not been able to 

 discover the Schwarzian derivative in either of them. 



The form as it stands .shows that for y a linear function of x and y may 

 lie substituted : and the form yeciprocated (by the interchange of .r and y) 



The consequence is that every pure reciprocant corre- 

 sponds to, and indicates, some singularity or characteristic 

 feature of a curve, and vice versa every such singularity 

 of a general nature and of a descriptive (although not 

 necessarily of a projective) kind, points to a pure reci- 

 procant. 



Such is not the case with mixed reciprocants. They 

 will not in general remain unaltered when linear substi- 

 tutions are impressed upon the variables. Is it then 

 necessary, it may be asked, to pay any attention to mixed 

 reciprocants ; or may they not be formally excluded at 

 the very threshold of the inquiry .' Were I disposed to 

 put the answer to this question on mere personal grounds, 

 I feel that I should be guilty of the blackest ingratitude, that 

 I should be kicking down the ladder by which I have risen 

 to my present commanding point of view, if I were to turn 

 my back on these humble mixed reciprocants, to which I 

 have reason to feel so deeply indebted ; for it was the 

 putting together of the two facts of the substantial per- 

 manence under linear substitutions impressed upon the 

 variables of the Schwarzian form and the simpler one 

 which marks the inflexions of a curve — it was, if I may so 

 say, the collision in my mind of these two facts — that 

 kindled the spark and fired the train which set my 

 imagination in a blaze by the light of which the whole 

 horizon of Reciprocants is now illumined. 



But it is not necessary for me to defend the retention 

 of mixed reciprocants on any such narrow ground of 

 personal predilection. The whole body of Reciprocants, 

 pure and mixed, form one complete system, a single 

 garment without rent or seam, a complex whole in which 

 al! the parts are inextricably interwoven with each other. 

 It is a living organism, the action of no part of which can 

 be thoroughly understood if dissevered from connection 

 with the rest. 



It was in fact by combining and interweaving mixed reci- 

 procants that I was led to the discovery of the pure binomial 

 reciprocant, which comes immediately after the trivial mono- 

 mial one, — the earliest with which I became acquainted, and 

 of the existence of compeers to which I was for some time 

 in doubt, and only became convinced of the fact after the 

 discovery of the Partial Differential Equation, the master- 

 key to this portion of the subject, which gives the means 

 of producing them ad libitum and ascertaining all that 

 exist of any prescribed type. Of this partial differential 

 equation I shall have occasion hereafter to speak ; but 

 this is not all, for, as we shall presently see, mixed reci- 

 procants are well worthy of study on their own account, 

 and lead to conclusions of the highest moment, whether 

 as regards their applications to geometry or to the theoiy 

 of transcendental functions and of ordinary differential 

 equations. 



The singularities of curves, ta'<ing the word in its 

 widest acceptation, may be divided into three classes : 

 those which are independent of homographic deformation 

 and which remain unaltered in any perspective picture of 

 the curve ; those which, having an express or tacit refer- 

 ence to the line at infinity, are not indelible under per- 

 spective projection, but using the word descriptive with 

 some little latitude may, in so far as they only involve a 

 reference to the line at infinity as a line, be said to be of a 

 purely descriptive character ; and, lastly, those which are 

 neither projective nor purely descriptive, having relation 

 to the points termed, in ordinary parlance, '■ circular 

 points at infinity '' [for which the proper name is " centres 

 of infinitely distant pencils of rays," i.e. pencils, every ray 

 of which is infinitely distant from every point external to 

 it]. Such, for instance, would be the character of points of 

 maximum or minimum curvature, which, as we shall see, 

 indicate, or are indicated by, that particular class of Mixed 

 to which I give the name of " Orthogonal Reciprocants." 

 All purely descriptive singularities alike, whether pro- 

 shows th It a similar substitution may be made for -v. Hence arbitrary 

 linear substitutions may be simultaneously impressed as x and y without 

 inducing any change of form. 



