404 



NA TURE 



[September i j, i8to 



The following is the list of grants to be submitted to 

 the meeting of the General Committee to-day :— 



A. — Mathematics and Physics. 



Seismological Phenomena of Japan 



Electrical Standards 



Meteorological Observations on Ben Nevis 



Electrolysis 



Photographs of Meteorological Phenomena 



Discharge of Electricity from Points 



Ultra- Violet Rays of Solar Spectrum 



Seasonal Variations of Temperature 



B. — Chemistry. 



Analysis of Iron and Steel 



Isomeric Naphthalene Derivatives 



Formation of Haloid Salts 



Action of Light upon Dyes 



C. — Geology. 



Erratic Blocks 



Fossil Phyllopoda 



The Geological Record 



Photographs of Geological Interest 



Lias Beds in Northamptonshire 



Registration of Type Specimens of British Fossils 



Volcanic Phenomena of Vesuvius 



Underground Waters 



Investigation of Elbolton Cave 



D. — Biology. 



Marine Biological Association at Plymouth 



Botanical Station at Peradeniya 



Improving Deep-sea Tow-net 



Disappearance of Native Plants ' 



Zoology of the Sandwich Islands 



Zoology and Botany of the West India Islands ... 



E. — Geography. 

 Normal Tribes of Asia Minor and Northern Persia 



G. — Mechanical Science. 

 Action of Waves and Currents in Estuaries 



H. — Anthropology. 



New Edition of "Anthropological Notes and Queries ' 



Anthropometric Laboratory 



North-western Tribes of C anada 



Habits of Natives of India 



Corresponding Societies 



SECTION A. 



MATHEMATICS AND PHYSICS. 



\o 



ICG 



SO 

 5 

 5 



lO 



SO 



20 



30 

 SO 

 40 



S 

 100 

 100 



30 



ISO 



50 



;^I330 



Opening Address by J. W. L. Glaisher, Sc.D., F.R.S., 

 President of the Section, 

 No one who is called upon to preside over this Section can 

 fail to be struck by the range of subjects comprehended within 

 its scope. The field assigned to us extends from the most exact 

 of all knowledge, the sciences of number, quantity, and position, 

 to branches of inquiry in which the progress has been so slight 

 that they still consist of little more than collections of observed 

 facts. This breadth of area has obvious disadvantages, but it is 

 not without some compensating advantages. In these days, 

 when science is so much subdivided, it is well that students of 

 subjects even so diverse as those with which we have to deal 

 should occasionally meet on common ground, and have the 

 opportunity of learning from each other's lips the kind of work 

 in which they are engaged. Wide as is our range, we should 

 remember also how closely knit together in various ways are the 

 more important of our subjects ; and in the case of mathematics, 



NO. 



1089, VOL. 42] 



astronomy, and physics, besides their actual and historical 

 alliance, a mathematician may be permitted to feel that a special 

 bond of union is created by the mathematical processes and 

 language which are essential for their investigation and ex- 

 pression. 



It is, I am afraid, unfortunate for my audience, that my own 

 subject should be at one extreme, not only of those dealt with 

 by our Section, but even of the still greater range covered by the 

 Association. I will endeavour, however, in my remarks to con- 

 fine myself to a few general considerations relating to pure 

 mathematics, which I hope will not be considered out of place 

 on this occasion. 



By pure mathematics I do not mean the ordinary processes of 

 algebra,_ differential and integral calculus, &c., which every 

 worker in the so-called mathematical sciences should have at his 

 command. I refer to the abstract sciences which do not rest 

 upon experiment in the ordinary sense of the term, their funda- 

 mental principles being derived from observations so simple as 

 to be more or less axiomatic. To this class belong the theories 

 of magnitude and position, the former including all that relates 

 to quantity, whether discrete or continuous, and the latter in- 

 cluding all branches of geometry. The science of continuous 

 magnitude is alone a vast region, containing many beautiful and 

 extensive mathematical theories. Among the more important 

 of these may be mentioned the theories of double and of 

 multiple periodicity, the treatment of functions of complex 

 variables, the transformation of algebraical expressions (modern 

 algebra), and the higher treatment of algebraical and ditterential 

 equations as distinguished from their mere solution. It is this 

 kind of scientific exploration which fascinates and rewards the 

 pure mathematician, and upon which his best work is most 

 profitably spent. I do not wish to under-estimate the import- 

 ance of such a subject as finite differences, in which a number 

 of distinct problems are treated with more or less success by 

 interesting methods specially adapted to their solution. Nor 

 would I willingly undervalue the interest of those branches of 

 mathematics which we owe to the mathematical necessities of 

 physical inquiry. But it always appears to me that there is a 

 certain perfection, and also a certain luxuriance and exuberance 

 in the pure sciences which have resulted from the unaided, and 

 I might almost say inspired genius of the greatest mathema- 

 ticians which is conspicuously absent from most of the investiga- 

 tions which have had their origin in the attempt to forge the 

 weapons required for research in the less abstract sciences. To 

 illustrate my meaning, I may take as an example of a subject of 

 the latter class the theory of Bessel's functions. The object of 

 mathematicians in this case has been to investigate the properties 

 of functions which have already presented themselves in astro- 

 nomy and physics. Formulae for their calculation by means of 

 series, continued fractions, definite integrals, &c., have been 

 obtained in profusion, numerous theorems of various kinds and 

 applicable to different purposes have been discovered, extensions 

 and developments have been made in all directions, and, finally, 

 the large body of interesting analysis thus accumulated has been 

 classified and systematized. But, valuable and suggestive as are 

 many of the results and processes, such a collection of facts and 

 investigations is necessarily fragmentary. We do not find the 

 easy flow or homogeneity of form which is characteristic of a 

 mathematical theory properly so called. In such a theory as, 

 for example, the theory of double periodicity (elliptic functions), 

 the subject develops itself naturally as it proceeds ; one group 

 of results leads spontaneously to another ; new and unexpected 

 prospects open of themselves ; ideas the most novel and striking, 

 which penetrate the mind with a charm of their own, spring 

 directly out of the subject itself. We are surprised by the 

 wonderful connections with other subjects which unexpectedly 

 start into existence, and by the widely different methods of 

 arriving at the same truths ; in fact, as our knowledge pro- 

 gresses, we continually find that results which seemed to lie far 

 away in the interior of the subject — so remote and concealed 

 that, at first sight, we might think that no other path except the 

 one actually pursued could have reached them — are actually 

 close to its edge when approached from another side, or viewed 

 from another stand-point. We notice, too, that any great 

 theory gives rise to its own special analysis or algebra, frequently 

 connecting together into one whole what were hitherto merely 

 isolated and apparently independent analytical results, and 

 affording a reason for their existence, and also — what is often 

 even more interesting — a reason for the non-occurrence of 

 others, which analogy might have led us to expect. I do not 



