NATURE 



617 



THURSDAY, OCTOBER 29, 1903. 



VECTORS AND ROTORS. 

 Vectors and Rotors, with Applications. By O. Henrici, 

 Ph.D., LL.D., F.R.S., and G. C. Turner, B.Sc. 

 Pp. XV + 204. (London : Edward Arnold, n.d.) 

 Price 4s. 6d. 



PROF. HENRICI can always be depended upon to 

 embellish any mathematical subject which he 

 touches, because, with the skill of the analyst, he 

 combines the keen perception of the geometer, which 

 ever seeks to render the results of analysis in some 

 way visible by spatial representation — or, perhaps, to 

 reach the results directly (and often more simply) with- 

 out any aid from analysis at all. To a mathematician 

 of this kind the subject of vector analysis is peculiarly 

 appropriate. We are therefore indebted to Mr. Turner 

 for putting into systematic form the lectures delivered 

 by Prof. Henrici at the City and Guilds Technical 

 College, and producing a very simple and elementary 

 work the methods and ideas of which should find a 

 very early introduction into our ordinary mathematical 

 teaching. 



The system here put forth is non-Hamiltonian. A 

 vector is throughout a mere " carrier." With Hamil- 

 ton it was this and more ; every unit vector, when em- 

 ployed as a factor, said Hamilton, is to be regarded 

 as a quadrantal versor the plane of which is perpen- 

 dicular to the vector. In the non-Hamiltonian system 

 the vector is not in any way associated with the notion 

 of rotation. Some vectors are, except as regards direc- 

 tion and sense, absolutely unrestricted in space ; others 

 (such as forces acting on a body) are restricted to 

 definite right lines and are called localised vectors. 

 For these latter the special name of " rotors " has 

 been invented, and Prof. Henrici must excuse an 

 adherent of the Hamiltonian system for saying that 

 this name seems to be wholly unjustified in a system 

 which refuses to associate the notion of a rotational 

 operation with any vector. Assuming that a " rotor " 

 means, perchance, a " rotator," how comes it that 

 such a name is applied to a mere " carrier "? There 

 is another term also adopted by Prof. Henrici the 

 justification of which is at least difficult, viz. the term 

 "ort. " A vector of unit length is called an "ort," 

 which is explained to be " short for orientation," and 

 " orientation " makes a dangerous suggestion of 

 rotation. The " ort " is, of course, Hamilton's unit 

 vector. The "rotor" and the "ort" should be re- 

 garded by anti-Hamiltonians as the trail of the 

 serpent.^ 



The contrast between the two systems is well illus- 

 trated by the discussion of the product, o/3, of two 

 vectors, o and $, which forms the subject of chapter iii. 

 of Prof. Henrici 's book. With Hamilton the nature 

 of the expression follows simply and naturally; o/3 

 means a//3-*, an operation implying rotation — the con- 

 version of the vector /3-' into the vector o. It can 

 therefore be taken as either a combined tensor and 

 versor operation, or a combined scalar and vector 

 operation. This at once gives us the complete specifi- 



1 Prof. A. Lodge suggests the term " locor " for rotor. 



Ho. 1774, VOL. 68] 



cation of the vector of aj3, and also that of the scalar 

 of oj8, making the latter equal to -ab cos 0, where a 

 and h are the tensors of o and 0, and e the angle 

 between them. 



Prof. Henrici, by a very simple and consistent rule, 

 specifies the vector part and makes it identical with 

 Hamilton's specification, but he makes the scalar 

 + ab cos 6, by what, after all, amounts to a perfectly 

 arbitrary and dogmatic definition (p. 95), its system- 

 atic connection with the mode of defining Vo0 being 

 somewhat strained and unconvincing. 



This, however, is a 'matter of no consequence, since 

 he is quite at liberty to lay down his own definitions^ 

 inasmuch as he is not hampered by the Hamiltonian 

 notion of rotation as associated with a vector. 



As regards notation in this part of the subject, it 

 may be pointed out that Prof. Henrici uses [o^] for 

 the Hamiltonian Vo/3, and (a,/3) instead of So/3, which 

 certainly does not seem to be an improvement, especi- 

 ally when we have to write down a long vector or 

 scalar equation — such, for example, as (iii.), p. 199. 

 Again, the notation [a|j3 + 7], instead of Va{fi + y), is 

 scarcely pleasing to the eye, even if it is not calculated 

 to lead to slips in working. 



The only indication that Prof. Henrici gives of his 

 view of the quaternion system is found in p. 104, where 

 he dispenses with the operation of division by vectors. 

 "This operation is complicated and will not be -con- 

 sidered at all. It leads to the much more complicated 

 Theory of Quaternions." It is, however, quite open to 

 a Hamiltonian to say nothing about division of vectors ; 

 he can treat his vectors as mere " carriers," and claim 

 all the results of a non-Hamiltonian theory as his 

 own ; for a non-Hamiltonian is not necessarily an anti- 

 Hamiltonian theory. It remains, of course, quite true 

 that with Hamilton division is the primary notion, and 

 multiplication the secondary. 



The subjects selected by Prof. Henrici for vector 

 treatment are geometrical and statical. Almost all 

 the prominent results of elementary geometry are 

 shortly and neatly obtained, and among the illustra- 

 tions of this subject are the Peaucellier and Hart 

 mechanisms for the description of a right line. There 

 is a very full discussion of centres of mass, and a plani- 

 nietric method of finding the centre of mass of any 

 area, which method is not so well known as it ought 

 to be. The determination of the centre of parallel 

 forces by the use of link (or funicular) polygons is fully 

 explained, while— to the great advantage of the student 

 —Prof. Henrici is very lavish of his figures. 



So very few elegances escape the watchful eye of 

 Prof. Henrici that one feels a pleasure in pointing out 

 something that he might have included in his dis- 

 cussion of force systems. The centre of a parallel 

 system of forces is known to everyone, but the astatic 

 centre of a system of coplanar forces has received little 

 attention. Yet it is a striking entity, and one which 

 is closely allied to the other centre. Its definition is 

 fairly well known ; perhaps the best specification of it 

 treats it as the point of intersection of the line of no 

 moment with the line of no virial. 



The portion of the book dealing with statics treats 

 largely of the stresses in frameworks, shearing forces, 

 bending moments, &c., the treatment being, of course, 



D D 



