TRANSACTIONS OF SECTION A. 561 



to inclose eacli product in brackets, and to indicate tbe nature of the products by 

 the kind of bracket used, viz., round brackets for the scalar-product, square brackets 

 for the vector-product. Thus, if small Greek letters denote vectors 



(a/3) = scalar-product, 

 [a/3] = vector-product. 



Then is 



(«^) = (/3a); [a/3]=-[/3al. 



If neither a nor /3 vanishes, then is 



(a/3) = the condition that a 1.3. 

 [a^] = „ „ „ a^/3. 



The product (aa) is denoted by a". 



If sums of vectors are tq be multiplied, the factors are separated by a vertical 

 line I . Thus the products of a + /3 into y are written (a + /3 | y), Sec. Then is 



(a + ^ I y) = (ay) + (/3y) ; [a + ^ | y] = [ay] + [/3y]. 



For these factors we have the products (n[/3y]) and [a[/3y]]. In the former the 

 law of association holds, and the square brackets may be left out, so that 



(a^y) = (a[/3y]) = ([a/3]y). 



This is the volume of the parallelepiped, with a, /3, y as edges. {a[3y) = is the 

 condition that a, (3, and y lie in a plane. 

 There is, besides, the formula 



[a[/3y]] = (ay)3-(a,3)y. 



These formulna contain the whole of the algebra of vectors as far as products 

 are concerned. Division may be altogether avoided. But it is sometimes con- 

 venient to introduce the reciprocal to a vector a, viz., by a~'- (not — ) is under- 



a 



«stood a vector of the same direction and sense as a, but of reciprocal length. 

 Then is 



a~i = --,, and {aa~^) = 1 . 

 a'~ 



The author adopts Oliver Heaviside's proposal of calling a vector whose magni- 

 tude or tensor is the ?uimberl,ai\ ' ort ' (from orientation). Hence if a is an 

 ort, then is a'- = l, not the laiit of le7i(/t/i, but the 7umibej- 1. 



The author also adopts Maxwell's right-handed sj-stem : ai3y, taken in cyclical 

 order, form a right-handed system if standing in a and looking towards /3, the 

 third vector y points to the left. The thumb, index-finger, and middle-finger, 

 when spread out so as not to lie in a plane, form on the right hand a right-handed 

 system, but on the left a left-handed. 



A right-handed system of three vectors mutually at right angles is called a 

 right system. 



A right system of ' orts ' he denotes generally by 'i, in, «3. The position 

 vector of a point is denoted bj^ p. If xgz are the rectangular coordinates (right- 

 Landed), then is 



p = (i-r + i.^y + 1.=. 



Here .ryz are lengths, not numbers. 



Another notation found convenient is in connection with Hamilton's differential 

 ■operator v (called nahla by Maxwell). Being of the nature of a vector, it com- 

 bines with a vector-function »;, according to scalar- or to vector-multiplication, 

 forming (v;) and [ v?]- In the former the brackets can often be left out. For 

 the latter it is convenient to use a special symbol, viz., v, with an arrow-head put 

 on top of it. As this requires a special type, formulae involving it are given on a 

 sheet reproduced from writing. This new symhol is called the vector-nabla. It 

 is a symbol for Maxwell's curl. 



1897, oo 



