164 PHILLIPS AND MOORE. 



An equation satisfied by (/> is then 



TT ((/)2 - 2 cos g,/ + 7) = (11) 



TT indicating the product of all the factors obtained by letting i, j 

 have the values in (8). For if .r is any vector in the plane kij, we have 



((^2 _ 2(/)COS(7i2 + 7)-.T = 

 and so 



TT (<^2 _ 2 '(j) COS qi2 + /) • .T = (12) 



Hence the left side of (11) is such that its product with any vector in 

 any one of the fixed planes of (j> is zero. Any vector in space can be 

 represented as a linear function of vectors in these fixed planes. Conse- 

 quently (12) is satisfied by any vector in space which proves relation 



(11). 



If some of the angles qij are equal, (10) can be replaced by the equa- 

 tion of lower degree obtained by using only one factor for each distinct 

 value of qij. For the above argument requires only one factor of each 

 type. If one of the angles is zero the resulting factor 



<|>2 - 2 + /=((/>- 7)2 



can be replaced by ^ — 7. For any vector x in the corresponding 

 plane is left invariant and so 



((f) — I) • X = X — X = 0. 



If one of the roots is tt, the factor 



c/>2 + 2 «A + 7 = ((/> + 7)2 



can be replaced by <^ + 7. 



The resulting equation containing only one quadratic factor for each 

 distinct value of qij, not more than one factor 4> — I and not more than 

 one factor cj) -\- I is the equation of lotvest degree satisfied by </>. For the 

 roots of <p are 



cos qa + V cos^gfj- — 1 = e^^' "^"^ • 

 and any equation satisfied by must be satisfied by all these roots. 



