168 PHILLIPS AND MOORE, 



Direct multiplication gives 



ii-My = :^qi,u-kiki-kjkj). 



Also 



1 ^^ ^ (A-'i ki "T" A'j kj). 



The characteristic equation of I-M is then 



7r[(J-M)2 + g,/7] = (19) 



TT representing the product for all values of qa. To obtain the equa- 

 tion of lowest degree satisfied by I-M we need only use one factor for 

 each distinct value of Qij and one factor I-M for each angle qij = 0. 

 This is shown as in the case of the equation of lowest degree satisfied 

 by (f) by taking the product with any vector in an invariant plane. 

 Hence the equation of lowest degree is of the same order as that of 0, 

 Furthemiore the values of qij (between — tt and + tt) are determined 

 by the roots 



of (j). Consequently if (18) is satisfied, Mi and M2, satisfy the same 

 equation of lowest degree (19) and the degree m of that equation is 

 the same as that of 0. By using (19), terms of degree higher than 

 m — 1 in 



can be replaced. Let the result be 



</, = an I + an (I-M{) + auil-MY +....+ ai,„(I-il/i)»-i 

 = an f + ai2 {I-M^} + an{I-My +....+ aUI-M^)^-^ 



(20) 



The coefficients an, 012, etc., are the same in both cases because we re- 

 duce the same analytical expression by the use of the same identity 



