WILSON AND LEWIS. — RELATIVITY. 407 



the plane may be written as r = .rp + yq,. By the postulated formal 

 laws, 



r.r = .r-p.p + ?/=q.q + 2.r//p.q. 



We may now note that by rotation a vector along a fixed line is con- 

 verted into a multiple of that vector. If p becomes 7?p, and the inner 

 product P'P remains invariant, then P'P = /rP'P; whence it is ob- 

 vious that P'P = 0. In general: The inner product of any singular 

 vector by itself is zero, and this suffices to characterize a singular 

 vector. Hence r*r reduces to 



r.r = 2.n/p.q. 



Before proceeding further with the definition of the inner product, 

 we may observe that the signs of x and y are determined by that one 

 of the four angles (made by the fixed lines) in which r lies. According, 

 then, as .r and y have the same sign or different signs, the vector r 

 belongs to one or the other of the classes (7) or (5), and the product 

 r-r will have one sign or the other. These considerations suffice to 

 show that if r and r' are two vectors, and if r^r and r'-r' have the same 

 sign, the vectors are of the same class, but if r^r and r'-r' are of op- 

 posite sign, r and r' are of different classes. We have here a marked 

 departure from Euclidean geometry, in which the inner product of a 

 real ^■ector by itself is always positive. 



We are now in a position to complete the definition of the inner 

 product by stating that the product is a scalar, and that the product 

 of a vector by itself is equal to the square of the interval of the vector, 

 taken positively if the vector is of class (7), negatively if of class (5). 

 This does not imply any dissymmetry between the classes (7) and (5), 

 but is only such a convention as is often made with respect to sign. 



The equation r-r = 2.vyP'q shows that the inner product of any 

 singular vector and any singular vector of the other class is equal to 

 one-half the inner product by itself of the diagonal of their parallelo- 

 gram. 



The inner product of any vector and a perpendicular vector is zcro.^^ 

 For by XVI it is evident that if p and q be the components along the 

 fixed directions of anj- vector r, so that r = p + q, then p — q is a 

 perpendicular \ector, and in general any perpendicular vector r' has 

 the form /?(p — q). Hence 



r'.r = /( (p — q)'(p + q) = n (p'P + p-q — q.p - q^q) = 0. 



17 The fact that the inner product of a singular vector by itself vanishes 

 justifies our convention that a singular line is perpendicular to itself. 



