76 



NA TURE 



[May 25, 189; 



Quaternions of Hamilton and Tail, the Amdehnunjslehre of 

 Grassmann, the vector analysis of Gibbs and Hsaviside. It is 

 this problem of how to harmonise, unify, generalise, and extend 

 that I have been studying. Analysts and physicists dislike Mr. 

 McAulay's idea of an independent plant ; they prefer to culti- 

 vate the old tree venerable with the growth of ages. 



After studying impartially all the writers at my command I 

 came to the conclusion that the analysis of vectors is comple- 

 mentary to the analysis of versors, and that the fundamentel 

 rules for the former are : — 



i"- = + >" = + /{•==+ ^ (I) 



ij = - ji j k = - kj ik = - ki (2) 



ij = k j k = i ki = j (3) ; 



whereas for the latter they are :-— 



II II II 



i-j'^ = - J- 



II II II 



,V^ = - H'' 



1'J' = - 



[ II II II u 



■ ^''J'= - j- k' 



II II II 



II 11 II I 

 i-' k- - - k- I'- 

 ll II II 

 k- i- = 



■ J- 



(4) 



(5) 



(6) 



It follows that in the manipulation of the products of vectors, 

 the distributive rule applies but not the associative ; while in 

 the products of versors both apply. These fundamental rules 

 for vectors are based on physical considerations, the principal 

 one of which is that the square of a vector is essentially positive, 

 whereas, according to quaternionists, it is essentially negative. 

 My view agrees with that principle of analysis which considers 

 the cosine in the first and fourth quadrants to be positive ; to 

 make it negative produces confusion and error. These principles 

 harmonise with those of Gibbs and Heaviside ; and in the memoir 

 quoted I have carried them out to their logical development. 

 It is this development which Prof. Knott characterises as " a 

 pseudo-quaternionic system of vector algebra, which is non- 

 associative in its products." I see no worthy aim in being 

 canny about the matter ; my sole aim was to develop the system 

 so that its truth or falsity might the more readily appear. At 

 the end of his article Prof. Knott admits that the assumflion 

 that the square of a unit vector is positive unity leads to an 

 algebra which is essentially different from the algebra of quater- 

 nions. As regards the fundamental principle being an assump- 

 tion, I refer him to that same chapter of " Kelland and Tait " 

 which he quotes, where he will find, italics and all : — " We 

 retain what Sir Wm. Hamilton terms the associative laws of 

 muiliplicalion : the law which assumes that it is indifferent in 

 what way operations are grouped, provided the order be not 

 changed ; the law which makes it indifferent whether we con- 

 sider a be Xohe a x b c ot a l> x c. This law is assumed to be 

 applicable to multiplication in its new aspect (for example that 

 i i k = ij.k) and being assumed it limits the science to certain 

 boundaries, and, along with other assumed laws, furnishes the 

 key to the interpretation of results. The law is by no means a 

 necessary law. Some new forms oflhe science may possibly 

 modify it hereafter. In the meantime the assumption of the law 

 fixes the limits of the science. " Here an authoritative expounder 

 places the quaternion algebra on precisely the same footing that 

 Dr. Knott places the "pseudo-quaternionic;" and he even 

 predicts that in the course of time such a complementary algebra 

 will be developed. It is incumbent on a critic, having admitted 

 the logical development, to show that the assumptions are ab- 

 surd, or correspond to nothing in physical science ; instead of 

 which he informs us that he is appalled by the complexity, but 

 nevertheless he feels sure that it contains nothing new. As 

 regards newness I invite his attention to pr. 95 of the "Prin- 

 ciples," where I have investigated the rules for the several 

 partial products of any number of vectors in space of 

 not more than four dimensions (and they may be 

 easily extended to space of higher dimensions). These consist 

 of certain rules of reduction which are to be taken along with 

 the rule of signs of determinants, thus embracing determinants 

 and Grassmann's combinatory products in the general theory of 

 products of vectors. He will also find there some reasons for 

 believing that the triad of rules No. 3 are very different in 

 nature from the other two triads, Nos. I and 2. It is possible 

 to get along without No. 3. 



That vectors should be treated vectorially, and versors ver- 

 sorially, and rotors rotorially, is neither nonsense nor a truism. 

 It is an important maxim, and of growing importance in tliese 

 days. Violation of it has produced the fundamental weakness 

 of Hamilton's analysis. In a more recent paper I have pub- i 



NO. 1230, VOL. 48] 



lished the generalisations for space of the exponential, binomial, 

 multinomial, and other fundamental theorems of analysis, and 

 I show that it was from treating versors vectorially that Hamilton 

 failed to discover them. 



Prof. Knott defines a quaternion as the quotient of two 

 vectors. Why choose the quotient ; is not the product always 

 the simpler idea ? But further on vectors are identified with 

 quadrantal quaternions, from which it follows that a quaternion 

 is the quotient of two quadrantal quaternions. I have devoted 

 some attention to logic ; but I fail to extract any meaning out 

 of this implicit definition. 



Prof. Knott informs the reader that whereas Heaviside and 

 myself find that v-« = d'-uldx- + druldy- ■\- d^iijil:? the real 

 V'-« is minus that quantity ; but he does not explain why Prof. 

 Tait prefers the unreal v'-u in his "Treatise on Natural 

 Philosophy." A scientific critic would, instead of using ex- 

 clamation points, proceed to show that in every case 

 V (Vm) = (VV) oi. If that can be proved, not from any fancied 

 properties of italic letters, but from physical considerations, 

 then I shall readily admit that v behaves as a versor rather than 

 a vector. The onus probandi lies on the minus men. 



Austin, Texas, May 6. Alexander Macfarlane. 



An Atmospheric Phenomenon in the North China Sea. 



During a recent wintry cruise in H.M.S. Caroline in the 

 North China Sea, a curious phenomenon was seen which may 

 be of interest to your readers. The ship was on passage 

 between Shanghai and the western entrance of the famous in- 

 land sea of Japan. On 24th February, at 10 p.m., when in 

 latitude 32° 58' N., longitude 126" 33' E., which, on reference 

 to the map, will be seen to be sixteen to seventeen miles south 

 of Quelpart island (south of the Korean peninsula) some unusual 

 lights were reported by the officer of the watch between the 

 ship and Mount Auckland, a mountain 6,000 feet high. It was 

 a windy, cold, moonlight night. My first impression was that 

 they were either some fires on shore, apparently higher from the 

 horizon than a ship's masthead, or some junk's "flare up" 

 lights raised by mirage. To the naked eye they appeared 

 sometimes as a mass ; at others, spread out in an irregular line, 

 and, being globular in form, they resembled Chinese lanterns 

 festooned between the masts of a lofty vessel. They bore north 

 (magnetic), and remained on that bearing until lost sigh: of 

 about midnight. As the ship was passing the land to the east- 

 ward at the rate of seven knots an hour, it soon became obvious 

 that the lights were not on the land, though observed with the 

 mountain behind them. 



On the following night, February 25th, about the same tine, 

 10 p.m., the ship having cleared Port Hamilton, was steeling 

 east, on the parallel of 34°, when these curious lights were 

 again observed on the same bearing, at an altitude of 3° or 4° 

 above the horizon. It was a clear, still, moonlight night, and 

 cold. On this occasion there was no land in sight on a north 

 bearing when the lights were first observed, but soon after- 

 wards a small islet was passed, which for the time eclipsed the 

 lights. As the ship steamed on at a rate of seven knots an 

 hour, the lights maintained a constant bearing (magnetic) of 

 N.2°W., as if carried by some vessel travelling in tlie same 

 direction and at the same speed. The globes of fire altered in 

 their formation as on the previous night, now in a mas.';ed 

 group, with an outlying light away to the right, then the 

 isolated one would disappear, and the others would take the 

 form of a crescent or diamond, or hang festoon-fashion in a 

 curved line. A clear reflection or gUare could be seen on the 

 horizon beneath the lights. Through a telescope the globes 

 appeared to be of a reddish colour, and to emit a thin smoke. 



I watched them for several hours, and could distinguish no 

 perceptible alteration in their bearing or altitude, the changes 

 occurring only in their relative formation, but each light 

 maintained its oval, globular form. 



They remained in sight from 10 p.m. until daylight (about 

 5.30 a.m.). When lost sight of the bearing was one or two 

 points to the westward of north. At daylight land 1300 feet 

 high was seen to the north and north-north-west, distant fifty 

 miles, the mirage being extraordinary. 



Thus, these lights were seen first in longitude 126° 33' E., and 

 last in longitude 128° 29' E. At first the land was behind 

 them, but during the greater part of the distance run it was 

 forty-five or fifty miles away to the north ; and the bearing of 

 the lights for at least three-fourths of the distance did not change. 

 On arrival at Kobe I read in a daily paper that the " Unknown 



