April 27, 191 1] 



NATURE 



2«I 



We thus arrive at an auxiliary, a;8, connected with the 

 fundamental notions by the relation, o^ = Saj3 + Vo;8, an 

 auxiliary the geometrical meaning of which will be seen 

 below. 



We then note the special case o^ = So-'= - (To)'^ so that 



-=:-—-;• With the help of this relation, we shall assign 



a Q 



meanings to S- , V- : 



V 



- S 



= V- 

 ./3 



From these we have S=+ V?= -^,% (S$a + Y0a) 

 a a la*' 



Ta- ■ 



a 

 Apart from this, the following geometrical consideration 

 will justify our introduction of the quotient and the con- 

 ception of an operator : — 



Tg.Tacos e Tg.T g sin 

 Ta2 '^'' ta^T 



where e is an unit vector +r\.o plane a, j3 

 _T/3cos0 T/3sin0 



OP PB OB 

 OA OA~OA~ 



Ta 



Wc have now, but not earlier, the conception of our 



auxiliary, whether the product o/3 or the quotient -, as 



a 

 an operator turning one vector into another, the former 

 ^-1 into a and the latter a into 13; this would, as usual, 

 justify our calling by their old name (quaternion) these 

 auxiliaries which we have here obtained from our funda- 

 mental conceptions, the scalar product and the vector 

 product. We may now proceed further and introduce into 

 Dur system the conception of the axes and the angle of a 

 quaternion. _ Wc may, and as a matter of fact shall, use 

 the quaternion whenever we find it expedient, but we must 

 not make it our fundamental notion. 



In view of the diversity of opinion shown above, some 

 modification and reconcilement on the lines suggested 

 above or on some other lines are absolutely necessary, if 

 the advocates of vector analysis are earnest in their desire 

 to see it universally applied, and the Cartesian and other 

 methods completely overthrown. 



Maumatha nath Ray. 



Calcutta Mathematical Society, Senate House, 

 March 2. 



lliis method of approach to the quaternion vector analysis 

 pr.ictirallv that rifloptoH bv Prnf. Joly in his " Manual 

 i Oii.iirrnlon'-. " 1 he iii.-ilinil is unsatisfactory, because it 

 ;:.ik<s ton i^i'i'.-it a (I'liiaiid ;it iho outset upon the learner's 

 ;l!i. Wliv shoiiM Sail he put equal to -^(i/mo^O? The 

 .-\'-'i' i--. "f (ciiisf, h. I .'lusf. that is the ^imphst way ol' 

 }4i'Uiiu'; a v ( Ini- al^( hra applicable to luiclidi-.aii space, and 

 at till' s.imc limi' assori.-ilivc in its vectni- [irochn ts. But 

 tiie exislciuc of so many varieties of non-assoeialive vector 

 algebra shows how absolutely unimportant this latter con- 

 sideration is to many who find vector .inaKsis useful. In 

 these varieties not only is th'if. no 'lion of 



a quantity a/3, where a and /J arc \ lO is a 



I>erfcct hatred of the mere suggestion of it is a quantitv 



worthy of general discussion, except (be it noted) in the 

 particular case in which a is perpendicular to j3. Mr. 

 Ray shows, by a simple Cartesian process, how easily we 

 may arrive at the recognition of this product if we start 

 with the geometrical definitions of Hamilton's Vaj8 and 

 Safi. But the method is unconvincing to the man who 

 prejudges the whole matter by barring out the quantity or 

 symbolic form a)3 as being fundamentally foreign to any 

 well-regulated system of vector analysis ! If they would 

 not listen to Hamilton, Tait, or Joly, will they listen to 

 any other quaternionist, charm he never so wisely? 



C. G. K. 



An Abnormal Zebra. 



In reference to the note by Prof. Ridgeway on a photo- 

 graph of a zebra, or boute-quagga, skin from the Athi 

 Plains of British East Africa, published in Nature of 

 April 20, I write to say that a copy of the same photo- 

 graph was received at this museum from Mr. Woosnam. 



As I have mentioned in a note in The Field of April 22, 

 Mr. Woosnam stated that there were only one or two of 

 such abnormally marked animals running in a herd of 

 granti at any one time. It is therefore clear that there is 

 no ground for regarding the variation as of racial value. 

 On this point Mr. O. Thomas, to whom the photograph 

 was sent by Mr. Woosnam, is in complete accord with 



myself. R. LVDEKKER. 



British Museum (Natural History), Cromwell Road, 

 London, S.W., April 24. 



A Robin and his Young. 



Last summer a pair of robins built their nest in an 

 old fish-basket that was hanging in a shed at the back 

 of my house. All went well until the young birds were 

 about a week old — then happened what appeared to me to 

 be a catastrophe. My Aberdeen terrier pup " Bebe," 

 who must have had some natural desire to catch the 

 mother bird, managed one morning to make a meal of 

 her. 



Contrary to what I should have expected, the male bird 

 kept close to his young family. Day by day I turned 

 oyer part of the garden to supply him with a little help in 

 his task. In due course he taught the whole of his young 

 family to fly. 



I have made inquiries, but cannot find anyone who has 

 had a similar experience, and wondered what your readers 

 might know about such cases. 



Charlie Woods. 



" Vectis," 2 Wellmeadow Road, I.ewisham, S.E., 

 April 18. 



NO. 2165, VOL. 86] 



PROPOSALS FOR THE REFORM OF TIIF 

 CALENDAR. 

 nrilE importance of a uniform and simple calendar 

 * is not a question which affords any f^round for 

 dispute. Whether regarded from the point of view 

 of the chronolopfist, striving to evolve order out of 

 regnal years and intercalary months, or fnMii that of 

 a business man in Cairo, transacting:;- alTairs with 

 clients who adhere severally to the Moslem, the Cop- 

 tic, the Hebrew, the Julian, and the Gregorian 

 calendars, the diversity of system from time to time, 

 from place to place, and between creed and creed, 

 is an exasperating and unmixed misforluno. The 

 Now Year festi\-,il is (•clchr.-ilcd hv the motlev races 

 which go to make up tiie |)<)i)ul.ilion of Siiii^aporo on 

 dates which extend over sevctjil munihs. In Con- 

 stantinople, until quite recently, even the division of 

 ilir (!.i\- was a source of grave inconvcnicnc(\ since 

 the (lav ended at local sunset. The peisivicnco ,,1 

 such anonialirs shows li()\> i]\,- \v,n,- ,,( (he 



reformer. Tradilinn and v ,;; idc, .md cMn 



the mere inertia ol ciisioni, arc gainst liini. 



From tlic poini of \icw ,,[ th- ■, .>ilil, a far 



greater advance would I>e inai' laiar step 



toward'^ the adonlion of one iii: ilcndai- than 



I)V nialcin" small llicui-ci ic.il ii iipi , .\ . i;i, nt s in a par- 



