482 TRANSACTIONS OF SECTION A. 



The following Papers were read : — 



1, On the Notation and Use of Vectors. 

 By Professor O. Henrici, F.R.S. 



2. The Notation and Use of Vectors, 

 By Professor C. G. Knott, D.Sc, F.R.S.E. 



It is undeniable that Hamilton's Quaternion Calculus embodies the first 

 completely worked out system of vector analysis ever given to the world. In 

 view of the excessive multiplication in recent times of vector notations, a question 

 worthy of consideration is, Do any of these notations excel Hamilton's in brevity, 

 in compactness, in graphic appeal to the eye and sense, in manipulativeness, and in 

 freedom from arbitrary invention of symbols P 



It is easy to show from the papers and books of most of the present-day non- 

 quaternionic vector analysts that they have not really studied Hamilton's method 

 at first hand. Had they done so they would liave seen that Hamilton's notations 

 are, in a manner familiar in the history of mathematical development, simply short- 

 hand expressions for the statements. Thus Vab is the vector of the product of 

 the vectors a, Zi ; Sab the scalar of the same product; and so on. These with 

 Hamilton are selective symbols, and pick out parts of the complete product of two, 

 three, four, or any number of vectors. These vector products are amenable to all 

 the ordinary rules of algebra with the exception of the commutative law. This 

 is not the case with the rival systems of vector algebra. For fancied convenience 

 at the start the associative law is disregarded. So long as we keep to products of 

 two vectors the associative law does not come into play ; hence for the simpler 

 applications all the systems are practically identical in their modes of attack on 

 any problem. But when products of three or more vectors or vector operators 

 have to be considered it is obvious that the calculus which retains the associative 

 law must have the greater flexibility. Especially is the value of the associative 

 law shown in the use of V, which in all non-quaternionic vector systems is robbed 

 of much of its power and analytical adaptability. 



It was the retention of the associative law which compelled Hamilton to 

 identify Sab with minus the product of the lengths of the vectors into the cosine 

 of the angle between them. The customary method is to use Hamilton's name, 

 but change the sign. But the change of sign involves far more than appears at 

 first. It compelled Gibbs to make use of a third kind of product in addition to the 

 so-called fundamental ones represented by Hamilton's —Sab and Vab; and more 

 recently it has compelled Jahncke also to introduce a third, but quite different 

 kind of product of two vectors in order to be able to treat of strains. This is 

 surely a confession of weakness. In the quaternion vector analysis as developed 

 by Hamilton and Tait, the treatment of the strain function grows naturally out of 

 the system. 



To compare the merits of notations, take the following expressions in Hamilton's, 

 Gibbs'Sj and Henrici's notations : — 



S.Ya^^ yd\,p, (ax(i).(iyx8)x{,xp)), ([a ^ ] [[y 8] [ep]]) 



or((axi3)(yK8)(ex/,)), or ([«^] [y Sj [f p]). 



Here, in addition to the six vectors which are common to all, Hamilton uses 

 eight other symbols of operation in the first case and five in the second, Gibbs 

 requires thirteen for the first and thirteen (or eleven) for the second, and Henrici 

 uses ten in the first and ten or eight in the second. Although the cross and dot 

 are the essential symbols in Qibb.s's notation, the brackets are absolutely necessary 

 in complex expressions, just as they are, but to a distinctly less extent, in the 

 quaternion notation. There is no do"ubt that Hamilton's is the more economical ; 

 it is in the natural order of the verbal description of each expression ; and it is 

 certainty not less graphic than either of the others. 



