28o 



NATURE 



[April 27, 191 1 



' ■' ..I...... I..... 1.... ,1.. not 81 all nupport an 



I . : I the aeroplane, which, 



it ia liupcd, will fi»e lioin aiiil ikitMnd Upon water with 

 lasc and perfect »afety. The flying fish, however, fre- 

 quently ttrtkcs a wave with one fin and is overturned, or 

 •trikeB it with violence. It would be very interesting to 

 know whether Belone does aid itself by its tail, and so it 

 in some way a parallel to the hydroplane boat. 



Cyril Crossuind. 



Dongonab, Port Sudan, Red Sea, March a^. 



The Stinging Tree of Pornaoaa. 



With reference to the letter on the Stinging Tree of 

 I orniosa in Naturji of March 3, it would be interesting 

 yuur correspondent would throw light on the exact 

 mechanism by which the sting in Laportea pterostigma 

 and L. crenulata is produced. L. cretiulata is locally 

 abundant in some parts of India. The curious point is 

 that the leaves are often glabrous. Moreover, the sting- 

 ing effects are, apparently, sometimes experienced without 

 actual contact with the plant. I was one day walking 

 through the hot, steaming forests near the Tista River, in 

 British Sikhim, with a friend. The Laportea was 

 abundant, and we carefully avoided it. On our way home 

 my friend was seized with the peculiar stinging sensations 

 of the Laportea in several parts of his body. These lasted 

 several days, and on the night immediately after being 

 stung became so bad that he was unable to get any rest 

 and became feverish. 



On another occasion I had to cut a survey line through 

 dense forest with an undergrowth of L. crenulata. The 

 coolies avoided the leaves as much as possible, and cut the 

 stems low. Some of them were stung on the body, but 

 all were attacked in different degrees with sneezing, violent 

 catarrh, and ultimately vertigo. I myself, although at 

 •ome distance from the actual cutting operations, though 

 I had to walk up the cut line, suffered to a less degree 

 in tho samp way. Yet I have often dashed a leaf across 

 the back of my hand with no ill effects! Sir J. Hooker 

 and others have noted that the effects are worse at some 

 times of the year than at others. The inflorescence, it 

 should be noted, is covered with hairs, and I have only 

 been able to account for the facts above described by sup- 

 posing that it is these deciduous hairs of the inflorescence 

 which get into the clothes and become inhaled when the 

 trre is shaken. H. H. Haines. 



Camp. Central Provinces, India, March 24. 



Fundamental Notions in Vector Analysis. 



I siiAi.i. be much obliged if you will kindly permit me. 

 through the columns of Nature, to make some suggestions 

 regarding fundamental conceptions in vector analysis, a 

 subject which was vigorously discussed in this journal 

 about twenty years ago (Nature, vols, xliii., xliv., xlvii., 

 xlviii., xlix.). The discussion showed that the slow pro- 

 gress of vector analysis was in a large measure due to the 

 want of unanimity as to its fundamental notions and 

 notations, and to an unfortunate aspect peculiar to it, viz., 

 a strong conviction on the part of the advocate of any one 

 of the various systems of vector analysis, that the other 

 systems, if allowed to grow, will do more harm than good, 

 while it may be noticed that in our ordinary scalar 

 analysis, although several systems {e.g. Cartesian, polar, 

 pedal, trilinear, &c.) exist side by side, there is no such 

 feeling. My object now is to suggest a system which, 

 while it aims at a reconcilement between the various 

 systems, will contain the best features of each of these 

 known systems. 



Dr. Knott (Nature, vol. xlvii., p. 590) justifies the 

 introduction of the quaternion as a fundamental conception 

 by saying that it is only a generalisation to the case of 

 vectors of the quotient (in the case of scalars) of two 

 lengths. But a great objection is that the quaternion — a 

 hvbrid conception, in part a scalar and in part a vector — 

 is not by itself capable of being defined in terms of the 

 three fundamental entities, magnitude, direction, and posi- 

 tion, as^ every fundamental conception oup:ht to be. No 

 such thing can, however, be said of the fundamental 

 notions of the non-quatemionists, the scalar product and 

 the vector product, which are defined in terms of only the 



NO. 2165, VOL. 86] 



fundamental notions of geometnr and trigonometry. I may 

 uUo repeat an argument of Prof. Gibbc (Nature, vol. xlvii., 

 p. 463) that the introduction of the acalar product and the 

 vector product as fundamental conceptions will meet Prof. 

 McAulay's observation {Ph. Mg., vol. xxxiii. 1893, p. 477) 

 that the arrest in the development of vector analysis is 

 due to the circumstance that quaternions are " independent 

 plants that require separate sowing and consequent careful 

 tending." Besides, as is pointed out b^ Prof. Gibbs 

 (Nature, vol. xliii., p. 511), it is not desirable that the 

 simpler conceptions should be expressed in terms of those 

 which are by no means so. It is not sufficient to say, as 

 has been argued ^Heaviside, Nature, vol. xlvii., p. 533), 

 that vector analysis should have a purely vectorial basis : 

 for that would only be a play of words. 



Now, although the non-quaternionists thus avoid certain 

 initial difficulties in presenting the subject, some of them, 

 viz., Mr. Heavisidc and Prof. Macfarlane, have made 

 innovations which not only have no justification, but have 

 created insuperable difficulties. We must have 0'= — 1, 

 and we must recognise the versional character of the 

 vector ; the principles of vector algebra must differ as little 

 as possible from the principles of scalar algebra, and we 

 cannot be blind to the usual meaning of equations such 

 as ij'=k, &c., as was pointed out by Dr. Knott (Nature, 

 vol. xlviii., p. 148: vol. xlvii., p. 590). All these diflS- 

 culties and others have arisen from an attempt to oust 

 the conception of a quaternion, whether in the initial or 

 at any later stage. So supreme is the contempt that 

 Gibbs, while dealing with the theory of dyadics, regards 

 a&-k-\ix+y9, a sum of expressions analogous to the 

 quaternions, as indeterminate, merely symbolic, having 

 physical meaning only when used as operator, although 

 scalars and vectors are derived from it. 



It is unfortunate that the advocates of vector analysis 

 cannot work in harmony with one another, recognising 

 superiority of each other in particular respects. Although 

 Gibbs admits that the quatcrnionic method has advantages 

 in certain cases, he would not tolerate its existence in the 

 field of vector analysis, or rely upon it in places where he 

 has found advantages. 



With regard to the question of notations, I may refer to 

 Nature, vol. xlvii., p. 590, where Dr. Knott rightly says 

 that the symbols used by the quaternionists for the scalar 

 product and the vector product express at once and clearly 

 the nature of the functions they represent, and that it is 

 not proper to use the sign of ordinary multiplication in a 

 case which docs not admit of one of the factors being 

 carried over to the other side as a divisor. 



I shall now work out the successive stages of introducing 

 the proposed system. We shall begin with the scalar 

 product, Saj8, and the vector product, VojB, defining the 

 former as a quantity equal to minus the product of the 

 length of one of the vectors, a, /3, and the projection on it 

 of the other, and the latter as a vector drawn perpendicular 

 to the plane of the vectors, of a a 



length equal to the area of -^1 



the parallelogram determined by / > 



them, so that rotation round it 

 from a to j8 through an angle 

 less than 180* is positive. We 

 see that we shall have 



So)8 = S/3o, Va/3= - V/3a. 

 Now if we take a = ix■^■k■jy■^+k^^ 



we have, Sft)8= -TaTj3 cos 



= - To X projection of T/3 on a 



Va3= » (projection of area of parm. a, on 

 .r plane) 

 + / (projection of area of parm. a, $ on 



y plane) 

 + j6 (projec'ion of area of parm. a, on 

 z plane) 



= (yi'\=-2 -^2=1 ) +Ah^i - 2?*1 ) + '^■^L>'S - J^S* 



.•. Sa/8+ Vo3= - [x^x^ f ^1 V2 + 2i=2> +'{J'\^-J'^) + 

 jiZyX^ - SaJTi ) + Mx^i - JTj ^l) 

 = ('-»^l -^m +*=!) ("^3+/>J + *Sl) 



