June 27, 1907] 



NA TURE 



201 



soon after migrated to Trinity, as his rival in the 

 same year ; while Tail and Steele were undergraduates 

 of the College, and Lord Kelvin (already Prof. W. 

 Thomson, of Glasgow) was a junior Fellow. 



Not long after taking his degree, in January, 1854, 

 being senior wrangler, and bracketed with Clerk 

 Ma.Kwell for the Smith's prizes, he began the career 

 of tuition of advanced honour men in mathematics, 

 which was soon to lead to a unique reputation as a 

 successful teacher. From 185S to iS8S he had, in all, 

 between 600 and 650 pupils, of whom the great 

 majority graduated as wranglers, twent\'-seven being 

 seniors, while forty-one were Smith's prizemen; be- 

 tween 1S61 and 1885, when he retired from this stren- 

 uous work at the age of fifty-four, he had all the senior 

 wranglers as pupils, with but one exception near the 

 end of the time.' The number of his pupils, which 

 was for many years about 100, was not at all unpre- 

 cedented; what was unique was the fact that for all 

 this time he directed, almost without challenge, most 

 of the intellectual activity of the elite of the under- 

 graduate mathematical side of the University. This 

 herculean task naturally demanded methodical ar- 

 rangements, and the husbanding of his resources to 

 the utmost. \Vhat he aimed at was to impart thorough 

 masterv of the main principles of ascertained know- 

 ledge over the field of mathematics then cultivated at 

 Cambridge ; it was clearly out of the question to 

 stray very far into the regions of nascent science in 

 which ordered theory gradually evolves itself in re- 

 sponse to concentrated and specialised effort. He was 

 in the habit of claiming that this would follow spon- 

 taneously in the case of the mathematician born, once 

 he had learnt mastery of the resources of the science ; 

 while even when it did not follow, the record in the 

 legal and other professions of persons who had done 

 well in youth in mathematical studies proved their 

 supreme value as a deductive mental discipline. 



His plan was to take small classes, each of about 

 ten men selected to run together, and to maintain 

 an average bv catechetical methods. Those who could 

 go faster than the average had e.xtra material pro- 

 vided in the form of manuscript digests for study, 

 and especially in the institution of a weekly paper of 

 about a dozen problems, selected from recent examin- 

 ation papers, or abstracted from memoirs in the home 

 and foreign mathematical jeurnals. An element of 

 competition formed a stimulus in answering these 

 papers, while written solutions were afterwards at 

 hand for study in cases of failure to unravel them. 

 Looking back on those times, it might be thought 

 that there was too much problem and too little sus- 

 tained theorv; but no one ever accused the standard 

 of the problems selected of being lower than it ought 

 to be, while, on the other hand, absence of some sijch 

 rigid procedure would have rendered quite impossible 

 that focussing of undergraduate mathematical ac- 

 tivitv and ambition in one place which was a main 

 feature of the system. Men with further ambitions 

 would struggle with Thomson and Tait's " Natural 

 Philosophy " or with Maxwell's " Electricity," or with 

 brilliant and stimulating courses of lectures given on 

 growing special subjects by the more eminent mathe- 

 matical physicis'? and thus learn that though in youth 

 masterv mav be .apid, yet at all times invention must 

 be slow. It was, moreover, thus possible for the abler 

 men to have time to spare to expand their outlook by 

 taking up some other branch of knowledge as a re- 

 laxation from mathematics, or for joining in other 

 activities of the L'niversity. Nowadays the field 

 covered by the mathematical instruction ofTered at 

 Cambridge is vastly wider than would have been con- 

 ceived as practicable twenty years ago ; but the ques- 



1 The-e ani other fact* hav \xm 'aken from a valuable notice in the 

 Cambrwi^ A-.f.V.i' ^igned W. VV. R. K. 



NO. 1965, VOL. 76] 



tion is still unsettled how far it is expedient to extend 

 the preliminary undergraduate course into complex 

 special theories. 



Whatever may be thought as regards Dr. Routh's 

 views on postponing special research in favour of 

 thorough preparation, it could not be urged that he 

 did not himself, notwithstanding his other absorbing 

 work, set an example of what research might be. 

 Many of his earlier papers, mainly in the Quarterly 

 Journal of Mathematics, related to the dynamics of 

 rigid solids, spinning tops, rolling globes, precession 

 and nutation, and such like, and were distinguished 

 by the development of methods relating to moving 

 systems of coordinate axes, and to the diflerentiation of 

 vectors such as velocity and momentum with regard 

 to them. In another connection he applied the kine- 

 matics of special systems of coordinate a.xes moving 

 along a curve to problems of curvature and torsion. 

 The advantages of these methods in difl'erential geo- 

 metrv have come again into recognition, as may be 

 seen in such works as Darboux's " Theorie des Sur- 

 faces." Afterwards, arising out of his researches 

 on dynamical stability, which will be referred to pre- 

 sently in more detail, there came a series of papers 

 in the Proceedings of the London Mathematical 

 .Society on the propagation of waves and the analysis 

 of complex vibrations in networks of interlacing 

 threads and in other such laminar systems, leading up 

 to a mechanical treatment or illustration of the broad 

 general theory of harmonic analysis, principal periods, 

 and related topics. 



In the early 'seventies, the question of the possible 

 explanation of steady, including apparently statical, 

 relations of material systems by the existence of latent 

 steady motions, such as the rotations of concealed fly- 

 wheels or gyrostats attached to the system, was inuch 

 to the fore. The fundamental problem as regards such 

 representations is their degree of permanence ; for a 

 state of motion which falls away, however slowly, 

 cannot be appealed to in elucidation of secular steadi- 

 ness of relations. .\t a later stacre the ideas of the 

 subject were crystallised by Lord Kelvin in his British 

 .Association address, Montreal, 1S84, entitled " Steps 

 towards a Kinetic Theory of Matter," and inlater ad- 

 dresses on cognate topics, mainly reprinted in vol. i. 

 (Constitution of Matter) of his " Popular Lectures and 

 .Addresses," culminating in a way in 1897 in his gyro- 

 static model of a rotationally elastic optical aether. 



It is thus not surprising that the .Adams prize sub- 

 ject at Cambridge for the period 1S75-7, announced 

 over the signatures of Challis, Clerk Maxwell, and 

 Stokes, should have been the search for " The 

 Criterion of Dynamical Stability." This subject 

 suited Routh's predilections exactly; and his classical 

 essay, " \ Treatise on the Stability of a Given State 

 of Motion, particularly Steady Motion," composed, as 

 he states in the preface, almost entirely during the 

 year 1876, was the result. The greater part of the 

 work in the essay is analytical, and is concerned with 

 the discussion of the nature of the roots of the alge- 

 braic equation determining the free period of slight 

 vibration of the dynamical system ; but where it enters 

 upon the discussion of dynamical principles, such as 

 the criteria connected with the Energy and the .Action, 

 the essay moves in a high plane. In particular, the 

 burning question of how adequately to represent latent, 

 and, therefore, unknown steady motions, such as those 

 of concealed flywheels or gyrostats attached to the 

 system, is solved at a stroke' by "the famous theorem 

 of the " modified Lagrangian function." It was estab- 

 lished, in fact, that the presence of concealed steady 

 motions does not fundamentally alter the standard 

 mode of analytical specification of dynamical inter- 

 action developed originally by Lagrange, except in 

 the one respect that the efTective Lagrangian function 



