230 



NA TURE 



{Jan. 7, 1886 



we all took a common interest, and in which it was a 

 matter of eager emulation between my hearers and myself 

 to try which could be first in at the death. 



During the past period of my professorship here, im- 

 perfectly acquainted with the usages and needs of the 

 University, I do not think that my labours have been 

 directed so profitably as they might have been either as 

 regards the prosecution of my own work or the good of 

 my hearers : my attention has been distracted between 

 theories waiting to be ushered into existence and provid- 

 ing for the daily bread of class-teaching. I hope that in 

 future I may be able to bring these two objects into closer 

 harmony and correlation, and think I shall best discharge 

 my duty to the University by selecting for the material of 

 my work in the class-room any subject on which my 

 thoughts may, for the time being, happen to be concen- 

 trated, not too alien to, or remote from, that which I am 

 appointed to teach ; and thus, by e.xample, give lessons 

 in the difficult art of mathematical thinking and reason- 

 ing — how to follow out familiar suggestions of analogy 

 till they broaden and deepen into a fertilising stream of 

 thought — how to discover errors and to repair them, 

 guided by faith in the e.xistence and unity of that intel- 

 lectual world which exists within us, and is at least as 

 real as that with which we are environed. 



The American Mathcinaticcil journal, conducted under 

 the auspices of the Johns Hopkins University, which has 

 gained and retains a leading position among the most 

 important of its class, whether measured by the value of 

 its contents or the estimation in which it is held by the 

 Mathematical world, bears as its motto — 



•iTpa-yixa.Twv ^Kei'xits oh 0\ewo^eywy. 



I have the pleasure of seeing among my audience this 

 day the most distinguished geometer of Holland, Prof. 

 Schoute, who has done me the signal honour of coming 

 over to England to be present at this lecture, who 

 hospitably entertained me at Groningen (in a vaca- 

 tion visit which I recently paid to his country, the classic 

 soil which has given birth to an Erasmus, a Grotius, a 

 Bcerhaave, a Spinoza, a Huyghens, and a Rembrandt), 

 and who was kind enough, in proposing my health at 

 a party where many of his colleagues were present, to say 

 that he felt sure " that I should return to England cheered 

 and invigorated, and would, ere long, light on some dis- 

 covery which would excite the wonder of the Mathematical 

 world." 



I do not venture to affirm, nor to think, that this 

 vaticination has been fulfilled in the terms in which it 

 was uttered, but can most truly say that the discovery, 

 which it has been my good fortune to be made the 

 medium of revealing, has excited my own deepest feel- 

 ings of ever-increasing wonder rising almost to awe, such 

 as must have come over the revellers who saw the hand- 

 writing start out more and more plainly on the wall, or 

 the scienziati crowding round the blurred palimpsest as 

 they began to be able to decipher characters and piece 

 together the sentences of the long lost and supposed irre- 

 coverable De Rfpublica. 



When I was at Utrecht, on my way to Groningen, 

 Mr. Grinwis, the Professor of Mathematics at that Uni- 

 versity, showed me an English book on " Differential 

 Equations," which had just appeared, of which he spoke 

 in high terms of praise, and said it contained over 800 

 examples. I wrote at once for the book to England, and 

 on seeing it on my arrival, forgetting that it had been 

 ordered, mistook it for a present from the author or 

 publisher, and, what is unusual with me, read regularly 

 into it, until I came to the section on Hyper-geometrical 

 series, where the Schwarzian Derivative (so named by 

 Cayley after Prof .Schwarz) is spoken of. 



Perhaps I ought to blush to own that it was new to me, 

 and my attention was riveted by the property it pos- 

 sesses, in common with the more simple form which 

 points to inflexions on curves, of remaining substantially 



unaltered, of persisting as a factor at least of its altered 

 self, when the variables which enter it are interchanged. 

 Following out this indication, I at once asked myself the 

 question, " ought there not to exist combinations of deri- 

 vatives of all orders possessing this property of recipro- 

 cation .'' " That question was soon answered, and the 

 universe of mixed reciprocants stood revealed before me. 

 These mixed reciprocants, by simple processes of com- 

 bination, led me to the discovery of the first pure reci- 

 procant, 3(5' - ^ac : whereupon I again put the question to 

 myself, "are there, or are there not, others of this form, 

 and if so, what are they ? " 



In an unexpected manner the question was answered, 

 and my curiosity gratified to the utmost by the discovery 

 of the partial differential equation which is the central 

 point of the theory, and at once discloses the parallelism 

 between it and the familiar doctrine of Invariants. Two 

 principal exponents of that doctrine, who have infused 

 new blood into it, and given it a fresh point of departure 

 — Capt. MacMahon and Mr. Hammond — I have the 

 pleasure of seeing before me. Mr. Kempe, who is also 

 present, has lately entered into and signally distinguished 

 himself in the same field, availing himself in so doing of 

 his profound insight into the subject of linkages, his 

 interest in which I believe I may say received its first im- 

 pulse from the lecture which he heard me deliver upon it 

 at the Royal Institution in January 1874, on the very night 

 when the Prime Minister for the time being sent round 

 letters to his supporters announcing his intention to dis- 

 solve Parliament. Of the two events I have ever regarded 

 the lecture as by far the more important to the permanent 

 interests of society. He has lately applied ideas founded 

 upon linkages to produce a most original and remarkable 

 scheme for explaining the nature of the whole pure body 

 of iSIathematical truth, under whatever different forms it 

 may be clothed, in a memoir which has been recommended 

 to be printed in the Transactions of the Royal Society, and 

 which, I think, cannot fail when published to e.xcite the 

 deepest interest ahke in the Mathematical and the Philo- 

 sophical worlds.' 



I also feel greatly honoured by the presence of Prof. 

 Greenhill, who will be known to many in this room from 

 his remarkable contributions to the theory of Hydro- 

 dynamics and Vortex Motion, and who has sufficient 

 candour and largeness of mind to be able to appreciate 

 researches of a different character from those in which 

 he has himself gained distinction. 



I should not do justice to my feelings if I did not 

 acknowledge my deep obligations to Mr. Hammond for 

 the assistance which he has rendered me, not only in 

 preparing this lecture which you have listened to with 

 such exemplary patience, but in developing the theory ; 1 

 am indebted to him for many valuable suggestions tend- 

 ing to enlarge its bounds, and believe have been saved, 

 b)' my conversations with him, from falling into some 

 serious errors of omission or oversight. Saving only our 

 Cayley (who, though younger than myself, is my spiritual 

 progenitor — who first opened my eyes and purged them 

 of dross so that they could see and accept the higher mys- 

 teries of our common Mathematical faith), there is no one 1 

 can think of with whom I ever have conversed, from my 

 intercourse with whom I have derived more benefit. It 

 would be an immense gain to Science, and to the best 

 interests of the University, if something could be done 

 to bring such men as Mr. Hammond (and, let me add, 

 Mr. Buckkeim, who ought never to have been allowed to 

 leave it) to come and live among us. I am sure that with 

 their endeavours added to my own and those of that 

 most able body of teachers and researchers with whom 

 I have the good fortune to be associated — my brother 

 Professors and the Tutorial Staff of the Universit) — we 



' In his memoir for the Phil. Trans. Mr. Kempe contends that any what 

 ever mathematical proposition or research is capable Of being represented by 

 some sort of simple or compound linkage. One would like to know by what 

 sort of linkage he would represent the substance of the memoir itself 



