ON THE USE OF VECTORIAL METHODS IN PHYSICS. 



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to about the notation used, and I have been asked to give a short account 

 of those now in use. 



Hamilton denotes vectors by small Greek letters. Maxwell changed 

 these to German capitals, and Heaviside these again to block letters. 

 Gibbs and likewise German authors use heavy type, the same letter in 

 ordinary type standing for the tensors. 



With regard to the notation of products greater divergency exists, 

 and besides the scalar-product of Hamilton differs in sign from that of 

 vector analysis. Keeping this in mind the following table will explain 

 itself : — 



I have used brackets for years and found them convenient. I was led 

 to them because I often found it confusing to decide at a glance how far 

 Hamilton's symbols S and V extended, and had to introduce brackets 

 to make this clear. ^ Professor Lorentz has adopted the same notatioTi. 

 Gibbs uses brackets [A B C] for the scalar-product of three vectors, which 

 otherwise would appear as A . B x C. 



There is a further product in quaternions which we may call the 

 quaternion-product. Hamilton denotes it by a j3 with 



or, in my notation, 



a/3 = Sa/3+V«/3, 



«/3=-(c,/3) + [«/3]. 



This gives rise to a product a ft y for Avhich the associative law holds, 

 and is a chief point in the theory of quaternions, the product being a 

 quaternion. 



Professor Joly in his letter shows that the theory of quaternions can 

 be based directly on this product by investigating the laws which make 

 the associative law true for n ft y. 



In this way a quaternion becomes defined by the product of two 

 vectors instead of by their quotient, and thus the theory of quaternions 

 can be much simplified. 



But if vector algebra has been studied independently of quaternions 

 then anyone who still wishes to study the latter can do so at once by aid 

 of the above equation as definition of a ftj, which is a quaternion. 



Note. — I learn from the January number of the ' Jahresbericht d. 

 Deutsch. Mathematiker-Vereinigung ' that on September 24, 1903, at 



* Hankel must have felt the same, for he writes Hamilton s products thus ; 

 S(aJ), V(aft), V[aVJc], &c. 



