April'?., 1886] 



NA TURE 



543 



is a result to have arrived at of the very first order of im- 

 portance. 



I have next in connection with that diagram to give another 

 whicli we owe to the labours of a German observer. Prof. 

 Sporer. Not only have we to accept the fact that these im- 

 portant solar phenomena are limited to certain zones, but we 

 have to study that fact in connection with another, that though 

 all of them vary very violently, they all have what is called a 

 cycle, and the cycle affects the particular zone of the sun on 

 which they appear. Here a sunspot curve, as it is called, writes 

 out for us in a graphic form the quantity of spotted area on the 

 sun from year to year. It begins at 1867, and ends at 1878. 

 This curve means that when the curve is at its highest, we get 

 the greatest number of spots, or the greatest amount of spotted 

 area on the sun's surface. At one place we get a very sudden 

 increase of the spotted area. The carve is almost like a chalk 

 cliff, it goes straight up, but it does not come so straight down. 

 The curve from the minimum of spots on the sun to the maximum 

 is very much steeper than that from the maximum to the next 

 minimum. The sunspot period on an average is one of about 

 eleven years, and it may be said, though I do not want the term 

 to be misunderstood, to represent the seasons on the sun, 

 because when we get that curve low, we see the sun for days 

 together without any spots on it at all. When we get to the 

 highest part, of course there is the greatest number of spots 

 on it. 



In connection with that period then, which, as it is good for 

 spots, must also be good for faculje and for metallic pro- 

 tuberances, after the results obtained by the Italian observers, 

 it is most interesting to see in further detail whetlier there is 

 any difference in the part of the sun thus aftected. The two 

 lower curves show us that when there is the smallest number 

 of spots on the sun — when there is a sunspot minimum — the spots 

 that appear are seen in a high latitude, and that latitude goes on 

 decreasing and decreasing regularly and gradually until v/e get, 

 at the next minimum, a real over-Lapping of two perfectly distinct 

 spotted areas. When we have the maximum period of sun- 

 spots, the latitude of the sunspot zone is between S" and lo*^, 

 but it gets much lower than that when the period is closing, and 

 even before one period has closed another one has begun in a 

 higher latitude, so that the swirl in the solar atmosphere seems 

 to begin in a high latitude — say 30° or 35°, or thereabouts — and 

 very soon gets into full swing in latitude between 10' and 

 12°, and then it very gradually dies away until spots and metallic 

 |>rominences and faculae cling pretty near to the equator, and 

 then we get a new wave of activity, beginning .again in a 

 high latitude, as is indicated by the beginning of the second 

 curve. 



Drawings made by Mr. Carrington a good many years ago 

 show this result in another form, which emphasises the enormous 

 difference in the amount of spotted area, as it is called, at the 

 maximum and minimum time. Another diagram gives the results 

 of the last eleven years' work at Greenwich, where they have 

 been computing the positions of the spots obtained on their 

 photographs and on the photographs which the Solar Physics 

 Committee receives from India. This gives the history of the 

 sunspot period in rather a different way ; we begin in the year 

 1873, and end in the year 1884, and the curves represent the 

 amount of faculse, of penumbrce, and umbrae. Here again we 

 get both faculfc, penumbr^e, and umbras increasing towards 

 the maximum period, and it is seen that when we come to dis- 

 cuss photographs instead of depending on eye-observations, as 

 the Italians did, we still find that the facula: and the spots 

 vary together. Another diagram shows another important 

 matter. We are now discussing at Kensington the results ob- 

 tained from the photographs from several points of view. One 

 point of view is this. It seemed hard, after all the trouble taken 

 to observe latitudes, that all spots north and south of the 

 equator should be lumped together in a mere statement of 

 spotted areas. The two upper curves in the diagram represent 

 the spots north and the spots south ; and an important thing 

 which comes out of this is that the curve representing the 

 greatest amount of spotted area north and that representing the 

 greatest amount of spot area north .and south do not go together. 

 We do not get the greatest amount of spots north and south 

 of the equator at the same time. A peak in the south curve is 

 in two or three cases associated with a valley in the north curve. 



J. Norman Lockyer 

 (To be continued.') 



THE CORRELATION OF THE DIFFERENT 

 BRANCHES OF ELEMENTARY AlATHE- 

 MA TICS^ 

 A MONG the permanent acquisitions to mathematical science 

 ■^"^ secured within the last half century, within the limits of 

 those branches with which our Association concerns itself, two 

 (I conceive) stand out as pre-eminent in their far-reaching and 

 all-pervading consequences. 



These are the firm establishment as distinct entities of two 

 concepts, which have been fixed for all the future of science in 

 the terms Energy and Vector, and the development of the groups 

 of ideas and principles which cluster around each. 



The term Energy indeed, and the great principle of the Co}i- 

 servatwn, or (as I prefer with H. Spencer to call it) Persistence, 

 of Energy, the establishment of which will live in the history of 

 science as the great achievement of the central part of the nine- 

 teenth century, h,ave a scope far beyond the purely mathe- 

 matical treatment of dynamics and the allied branches of 

 physical sciences. They, the concept and the principle, have 

 already profoundly modified the views of the physicist as to the 

 natural laws with which he is concerned, and are destined to 

 form the starting-point and firm foundation for all his conquests 

 in the future. But no le-s is it true that the conception of 

 energy, while it has naturally arisen out of the higher mathe- 

 matical treatment of dynamics, has necessitated a very material 

 recasting of that treatment in its most elementary, as well as in 

 its more advanced, stages, if it is to bear any fruitful relation to 

 physical science in general. This recasting of elementary 

 dynamics, if not yet fully and satisfactorily effected in most of 

 the text-books which still remain in use, in which the notion of 

 energy is brought in rather as the "purple patch" than in- 

 woven into the whole texture of the robe in which the subject is 

 clothed, is yet, thanks pre-eminently to the teaching of Max- 

 well, Thomson and Tait, and Clitford, in a fair way for being 

 accomplished. 



The influence of the conception of energy is, however, as 

 regards mathematics, rather an influence from without than one 

 from within its peculiar domain. 



That which is strictly mathematical in the treatment of any 

 science is not its subject-matter, but the form in which that sub- 

 ject-matter must from its nature be expressed. Mathematics, as 

 such, is in fact a formal (may I not say tlie formal ?) science, 

 concerning itself with the particular matter only so far as that 

 matter necessitates a particular form for its expression. Hence 

 the recurrence of the same formulce and mathematically the 

 same propositions in different branches of science, so that, to 

 take elementary instances, a proposition in geometry may be 

 read off as a proposition in statics by substituting forces for lines, 

 or the formula which deteimines the speed of the centre of mass 

 of two masses having difl^erent speeds is also that which deter- 

 mines the temperature resulting from the mixture of two masses 

 of different temperatures. 



To this formal, or essentially mathematical, part of the exact 

 sciences belongs the conception of a Vector, or rather the group 

 of conceptions which cluster around that term. The term itself 

 was introduced by Hamilton in connection with his grand theory 

 of quaternions about forty years since, but the idea had been 

 already firmly grasped and developed so as to afford a complete 

 explanation of the imaginary {\l - i) of ordinary algebra within 

 the twenty years preceding that epoch. In fact in the year 

 1845 I myself enjoyed the privilege, as a young student, of 

 attending lectures of De Morgan on this subject, which he after- 

 wards developed in his treatise on "Double Algebra," published 

 in 1849.- I think, however, that we may conveniently date 

 from the introduction of the term " Fiv/or," which is now the 

 accepted term for any magnitude which besides numerical 

 quantity or intensity has a definite direction in space, the defini- 

 tive acquisition of this concept with all its consequences to the 

 settled territory of mathematical science. The calculus of 

 quaternions indeed, or that part of it which was truly original 

 and due to the genius of Hamilton alone, involving the concep- 

 tions of the products and quotients of vectors in three-dimen- 

 sional space, is doubtless beyond the range of what now can be, 

 or within the near future is likely to be, regarded as elementary 

 mathematics ; but the notions of vector addition and subtraction 



^ A paper read before the Association for the Improvement of Geometrical 

 Teaching by the President, R. B. Hayward, F.R.S. (see l^AruRE for 

 January 21, p. 277). We print the address in the hope that a discussion of 

 some of its principles may ensue. 



■-■ Sir \V. R. Hamilton's Lectures on Quaternions were published in 1853. 



