July i6, 1903] 



NA TURE 



259 



period, it will be seen that a similar connection seems to 

 exist between the latitudes of the centres of action of the 

 prominences and the three types of coronas. 



The investigation seems to indicate that it is the sum 

 total of prominence action in the different zones which pro- 

 <luces the largest coronal streamers, and not any particular 

 prominence at any particular moment ; it is for this reason 

 ihat the form of the corona is not a fleeting phenomenon 

 changing every minute or hour, but one lasting over several 

 months, and sometimes as much as a year or more. That 

 the general form of the corona does undergo comparatively 

 slow changes is borne out, to a great extent, by the simi- 

 Jarity of coronas which are observed at eclipses which occur 

 tlose together, such as those in 1900, 1901, the two eclipses 

 jn 1889, &c. 



It is of great interest briefly to note the connection 

 between the centres of prominence action when either two 

 or one of them exist in each hemisphere. In the first place 

 -a well-defined large coronal streamer apparently origin- 

 ates, as many photographs indicate, not from disturbance 

 at the centre of its base, but near the two ends. Such a 

 streamer is generally made up of groups of incurving 

 structure, termed previously " synclinal " groups, and this 

 structure is, in many cases, very distinct. When there are 

 tiL'o centres of prominence action in one hemisphere, the 

 coronal disturbances resulting from each trend towards each 

 other, and constitute a large streamer with an. apparent 

 ^' arch " formation. If the two centres of prominence 

 action exist in comparatively mid-latitudes, one large 

 streamer is formed in each quadrant, and the form of the 

 corona is of the " intermediate " or " square " type. 



When one of the centres is near the region of the poles 

 and the other in comparatively low latitudes, the tendency 

 Is still for the two disturbed coronal regions to trend to- 

 wards each other, but they constitute either a large streamer 

 of an " arch " formation nearer the solar poles with a very 

 extended base, or two separate streamers which combined 

 have a fish-tail appearance. 



With one centre of action of prominences in each hemi- 

 sphere, the resulting coronal disturbances in both hemi- 

 spheres curve towards the solar equator, and form 

 apparently a large equatorial streamer; the "equatorial" 

 type of corona is here formed. 



The accompanying sketches (Fig. 2) illustrate in dia- 

 grammatic form the general relationships between the lati- 

 tudes of the spot zones, the latitudes of the centres of action 

 of the prominences, and the suggested resulting positions 

 and origin of structure of the coronal streamers for each 

 of the three types of coronas here discussed. It will be 

 noticed that in the case of the " polar " and " inter- 

 mediate " types, when the sun-spots are numerous, the 

 zones in which they occur have apparently little connection 

 with the coronal streamers. When the latitudes of the 

 spot zones do approximate more nearly to the bases of the 

 coronal streamers, as in the " equatorial " type, and might 

 be considered as being the origin of their existence, the 

 spots at these epochs are near a minimum, that is, are very 

 few and small in size, and have the least power of action. 

 William J. S. Lockyer. 



SOME PRESENT AIMS AND PROSPECTS OF 

 MATHEMATICAL RESEARCH."^ 



TT may be doubted on the whole whether any completely 

 scientific and permanent dividing lines for the classifi- 



ation of modern original work of pure and applied mathe- 

 matics can be drawn. 



The nearest approach is perhaps an arrangement accord- 

 ing to motive. Thus a first class may be constituted of 

 those investigations which aim at discovering and establish- 

 ing the foundations of the subject, and obtaining rigorous 

 proofs of theorems already known ; such work as that 

 which Peano and Russell are doing in their symbolic nota- 

 tion for the general principles of mathematics, or Fieri and 

 \'eronese for the axioms of geometry, or Picard for the 

 existence theorems of differential equations, or Vall^e- 

 Poussin for the differentiation of definite integrals. 



1 From an address by Mr. E. T. Whittaker on " Some Present Aims and 

 Prospects of Mathematical Research," delivered before the University 

 College Mathematical Society on June 25. | 



NO. 1759, VOL. 68] 



Although the primary aim of such papers is that of im 

 parting a strict logical rigour to the theory discussed, yet 

 the most surprising and unexpected new results are con- 

 stantly arising in them ; as an instance, I may mention 

 Fano's discovery of a space which consists only of 15 points, 

 and which satisfies all the conditions for an ordinary pro- 

 jective space except the condition that each part is to be 

 distinct from its harmonic conjugate ; or the remarkable 

 result that a projective geometry of two dimensions cannot 

 be obtained without the supposition that the two- 

 dimensional space is contained in a three-dimensional space ; 

 or the well-known theory of Fourier series and integrals 

 which can represent different analytic functions in different 

 parts of their domain of existence. It is a notable fact that 

 this type of research seems peculiarly congenial to the mind 

 of the Latin races. Undoubtedly much work of the kind 

 has been done in Germany during the nineteenth century, 

 but the honour of its foundation must be assigned to 

 Cauchy, and its home has always been in France and Italy. 

 In this country it has never thoroughly taken root, perhaps 

 because, as someone said, the Englishman cannot dis- 

 tinguish between a proof and an appeal to the jury. In 

 America, however, a considerable amount of attention is 

 now given to the subject by such writers as Moore, Osgood, 

 Bdcher, and Huntington. 



A second class of research can be formed from those 

 which are directly provoked by some observed phenomenon 

 of nature, researches of which the immortal type is New- 

 ton's discovery that if the planets move in ellipses with 

 the sun in one focus, it must be because they are attracted 

 to the sun with a force which varies as the inverse square 

 of the distance. 



In work of this kind our country has always borne a 

 distinguished share ; the greatest achievements of the 

 English school of mathematical physicists must all be 

 included in it, and even at the present time no paper excites 

 so much interest among us as one which gives a mathe- 

 matical explanation of the Zeeman effect or the second law 

 of thermodynamics. 



A third class of investigations may be made to consist of 

 those in which the motive is not in some external 

 phenomenon, but in what may be called the internal ex- 

 pansive force of the subject itself, the inherent capability 

 of extension, which is latent in every theorem of mathe- 

 matics, the desire of the mathematician who has solved the 

 quadratic equation to solve the cubic and quartic, and then 

 either to solve the quintic or to show that it cannot be 

 solved by radicals. 



This, which is by far the largest of the three classes, 

 admits of several subdivisions, according as the successful 

 issue of the work is due mainly to the author's geometrical 

 imagination, as in the writings of Cremona and Chasles, 

 or to his power of algebraical analysis, as in much of the 

 work of Jacobi and Cayley, or to his having brought to 

 bear on the subject a novel set of ideas, as, for instance, 

 in Fuchs's papers on linear differential equations, or to what 

 may be called pure constructive intuition, which does not 

 depend on the extension and generalisation of preceding 

 results, as for instance, Euler's expression for the gamma- 

 function as an infinite product, or his solution of the many 

 types of differential equations. 



The second of these subclasses, namely, that in which 

 the successful management of highly complicated symbolic 

 work is the most prominent feature, has flourished perhaps 

 more than any other branch of non-physical mathematics 

 in our own country. 



It may be questioned whether this is not in part a con- 

 sequence of the traditional English mode of training, which 

 includes far more working of hard examples than is 

 customary abroad, and thereby gives the mathematician 

 that algebraical power which comes of much practice : but 

 no one can see such work as that of Cayley or Forsyth 

 without feeling that it is largely due to an inherent 

 algebraic power with which our distinguished fellow- 

 countrymen have been endowed. The introduction of new 

 algorithms and new concepts is, on the other hand, a 

 German characteristic ; a notable instance is furnished by 

 the invariant-theory, which, after its first development by 

 Cayley and Salmon on purely algebraical lines, was trans- 

 formed by Aronhold's introduction of the symbolic nota- 

 tion. The Mengenlehre of Cantor, the Ausdehnungs- 

 lehre of Grassmann, numerative geometry and the theory 



