October 5, 1893] 



NATURE 



54 r 



that I advocate nothing of the sort. What I do advocate is to 

 treat vectors as vectors, and versors as versors, and I show that 

 the products of versors differ essentially from the products of 

 vectors in that the associative rule applies to the former, but 

 not to the latter. Prof. Knott justifies the treatment of quad- 

 rantal versors as vectors, because they are compounded accord- 

 ing to the parallelogram law. It is true that the components of 

 a quadrantal versor are so compounded, because every versor 

 involves an axis ; but the minus comes in, not on account of the 

 axis, but on account of the angle of the versor, the very element 

 which differentiates it from a vector. 



I have said that v" = 7^ + - . + -,- is more consistent with 

 ax- ay ■ dz- 



analysis than v= = -( — +™ + --Y and I have remarked that 



\dx' ay- di'l 

 in works on mathematical physics, even in Kelvin and Tail's 

 "Natural Philosophy," the minus was dropped. A sign that 

 can be so readily dropped has probably got no good reason for 

 its appearance. In reply. Prof. Knott says that " when V'-f 

 occurs in ordinary non-quaternion analysis, it is used in the 

 sense of the tensor, for only as such can it come in." This ex- 

 planation does not explain ; for " the name tensor is applied to 

 the posithe number which represents the length of a line " 

 (" Hamilton's Elements," p. 164). Now the ordinary analysis 

 is not limited to signless quantities, but embraces quantities 

 which may be positive or negative. Why then is the minus 

 dropped in an analysis where sign is essential ? I asked for 

 a proof of the principle that v(Va)) = v-<o ; it is replied that 

 " in quaternions there is no doubt whatever." Are we per- 

 mitted, then, to doubt it as a truth in ordinary analysis, being 

 •true only in quaternions? If it is a matter of convention, no 

 one desires two contradictory systems of analysis ; if it is a 

 matter of truth , it cannot be true " in quaternions " and not 

 in ordinary analysis. 



I have said that the rule ij = k expresses what is true in 

 space of three dimensions. Prof. Knott asks: "If a vector 

 cannot be a versor in product combinations, what is the signifi- 

 cation of the equation ij = k .? " Let us first of all remove every 

 ambiguity from the equation. We have then in all three 

 cases : first, i and J botn quadrantal versors ; second, J a versor 

 and j a vector ; third, i and j both vectors. To distinguish 

 between a quadrantal versor and a vector, let the former be 



denoted by i'- . Thenz'^y- = - k' means the forward order 

 being taken, that a quadrant round i followed by a qudrant 

 round j is equivalent to a quadrant round the opposite of k. 



ir 



Again, ij = k means that the vector y, when turned through a 

 quadrant round t coincides with k. Finally, ij means the unit 

 of directed area which has i for base and/ for altitude ; for 

 some purposes it may be represented by k on the principle that 

 the axis of a plane may be specified by the axis which it wants ; 

 but at p. 92 of " The Principles of the Algebra of Physics," 1 

 have shown that the several types of products of vectors maybe 

 formed independently of that principle. Prof Knott states that 

 he fails to see what physical considerations have to do with 

 mathematics of the fourth dimension. It is evident, however, 

 that his perception cannot be taken as a criterion of truth, for 

 every type of product of four vectors is geometrically real 

 excepting the one which supposes them all independent of one 

 another. 



I have said that the rule? for differentiation are much simpli- 

 fied when vectors and versors are not confounded. In proof of 

 this I invite comparison. 



I have said that the principles of quaternions can be greatly 

 extended. In my papers will be found for the first time the 

 extension of space analysis to logmarithmic spirals and to hyper- 

 bolic trigonometry. The connection of the latter with non- 

 euclidean geometry is also pointed ou'. As further evidence of 

 the fruitfulness of my notation and principles I may mention 

 that I have just read before the Mathematical Congress assembled 

 at Chicago two papers^one on "The Definitions of the Trigo- 

 nometric Functions," the other on " The Principles of Elliptic 

 and Hyperbolic Analysis." These papers give the trigono- 

 metry of^the elliptic and' hyperbolic surfaces. 



As regards Prof Knott's closing quotation from " Paradise 

 Lost," I feel like the .Senior Wrangler who, having read through 

 the poem, remarked that it was all very pretty, but he didn't 

 quite see what it proved. I close with a quotation which is 



from as good a book, and possesses more logical force: "Ye 

 shall know them by their fruits. Do men gather grapes of 

 thorns, or figs of thistles?" Alexander Maci-arlane. 



Chicago, 111., August 26. 



Astronomical Photography. 



The letter from Lord Rayleigh in your issue of August 24, 

 on the subject of "Astronomical Photography," will, it is to 

 be hoped, elicit some information from photographic experts. 



Meanwhile, accepting what Lord Rayleigh says as to the 

 present possibilities in the preparation of plates, I fail to see 

 where any considerable saving is to be effected in the cost of the 

 apparatus, as he appears to suggest. 



For astronomical photography a pair of telescopes are re- 

 quired. The larger of these is employed to take the photo- 

 graphs, and the smaller acts as a'guider. Supposing that plates 

 could be obtained which were acted upon by visual rays, while 

 cnmparatively insensible to the violet and ultraviolet light, 

 this would simply mean that both the objectives would have to 

 be made visually perfect, instead of having one of them as here- 

 tofore corrected for violet and ultra-violet light. A photo- 

 graphic objective is no more costly than a visual one of the same 

 aperture ; and as to mounting clockwork and dome, there could 

 be no difference in expense. 



Of course, if the necessity for a separate guiding telescope 

 could be avoided by the adoption of Lord Rayleigh's sugges- 

 tion, there would in general be some saving of expense ; it 

 should, however, be noted, that even when reflectors are em- 

 ployed for taking the photographs, it has not been always found 

 desirable to dispense with the guiding telescope, though in this 

 case, of course, the question as to the nature of the plates 

 cannot ari^e at all. 



In the particular instance of the instrument now proposed for 

 Cambridge, the guiding telescope is already to hand in the 

 shape of the present Northumberland instrument. 



It is certainly easier to test the qualities of an objective cor- 

 rected for visual rays than for photographic rays (if I may still 

 use language which Lord Rayleigh has pointed out as incorrect). 

 On this account it would, therefore, be desirable to have plates 

 such as he refers to, rendered available for astronomers engaged 

 in photographic work. Robert S. Ball. 



Observatory, Cambridge, September 12. 



P.S. — Sir Gabriel Stokes, after reading the above, writes: 

 " I would ask whether in an orthochromatic plate the blue and 

 violet are impressed more feebly than the rays which are 

 visually the brightest. It may be so, but I do not happen to 

 know whether it is." 



The Constellations of the Far East. 



With regard to the questions asked by " M. A. B." about the 

 grouping of stars into constellations (Nature, August 17), I 

 venture to answer the last two, which the limited knowledge 

 of an Oriental may partly meet, hoping thereby to interest 

 some of your readers. 



I do not consider that each race necessarily relies on its 

 own plan in the fabrication of constellations. The Coreansand 

 Anamese are said to be still adhering to the Chinese system, 

 and till lately the Japanese were doing so. It is strange to find 

 the latter, replete with so peculiar mythology, on which the 

 national claim for high ancestry rests, possessing very few 

 vernacular constellations. 



Undoubtedly the Chinese system is of peculiar aspect. A 

 name is given to a " Seat," which is sometimes a single star, but 

 in general a group of stars, varying in number from two to 

 twenty or thirty ; and in one group, the Imperial Body- 

 guards, they amount to forty- five. Occasionally the same stars 

 are at once named collectively and individually ; thus, the first 

 seven stars of Ursa .Major are grouped into Peh-tau or the 

 North Ladle, of which the scoop consists of Shu a, Siuen ;8, 

 Ki 7, and Kiuen 5, and the handle of Ynh-hang e, Kai-yang f, 

 and Yau-Kwang 7). With Polaris as the centre, the heavens 

 are radiantly divided into the twenty-eight "Inns" of unequal 

 breadths, each division being denominated after its typical con- 

 stellation, besides enclosing numerous Seats subordinate to the 

 latter. 



The fundamental idea of the plan is enigmatically expressed 

 thus : " Sing (the star) is Tsing (the spirit)." Its solution con- 



NO. 1249, VOL. 48] 



