COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 



335 



ind the resulting equation # 



1 cos(a + /3) cos a [ 



cos(a + /3) 1 cos/3 | = 0. 



cos a cos/3 1 



The same identity almost suffices to give the corresponding relation between 

 jin (a + j8), sin a, sin /8, the set of equations now being 



— sin /3 . cos a — sin a . cos /3 + sin (a + 6) = ' 



sin/3.cos(a + /3) - sin(a + /3) . cos/3 + sin a =0 



sina.cos(a + /3) - sin (a + /3). cos a + sin/3 =0 



sin (a+/3)- cos (a + /3) — sina.cosa — sin/3. cos/3 + 2 sina sin/3sin (a + /3) = 



vvhence on the elimination of cos a, cos fi, cos (a + /3) we have 



sin/3 sina sin(a + /3) 



sin fi . sin (a + |6) sina 



sina sin(a + /3) sin /3 



sin(a+/3) sina sin/3 2 sin a sin /3 sin (a +/3) 



= 



15. The consideration of the relation between cos (a + fi + y + S) and. cos a, cos/3, 

 job 7, cos $ leads at once to the question of the rationalization of the equation 



a + b Jxy + c Jxz + d Jxw + e Jyz + f Jyw + g Jzw + h Jxyzw = 0, 

 jecause 



cos(a + /3 + y + (S) = cos a cos /3 cos y cos 8 — 2cos y cos S sin a sin /3 + sin a sin /3 sin y sin 5 . 



By proceeding in exactly the same manner as in section 1 2 the result of the rationaliza- 

 tion is obtained in three forms, viz., (1) the product 



(a + b Jxy + c Jxz + d Jxw + e Jyz + f Jyw + g Jzw + h Jxyzw) 

 .(a + b Jxy + c Jxz + d Jxw — e Jyz — f Jyw — g Jzw - h Jxyzw) 

 .(a + b Jxy — c Jxz — d Jxw + e Jyz + f Jyw — g Jzw — h Jxyzw) 

 .(a + b Jxy - c Jxz - d Jxw — e Jyz - f Jyw + g Jzw -f- h Jxyzw) 

 .(a - b Jxy + c Jxz - d Jxw + e Jyz - f Jyw + g Jzw - h Jxyzw) 

 .(a - b Jxy + c Jxz - d Jxw - e Jyz + / jyw - g Jzw + h Jxyzw) 

 .(a - b Jxy — c Jxz + d Jxw + e Jyz — f Jyw - g Jzw + h Jxyzw) 

 .(a - b Jxy - c Jxz + d Jxw - e Jyz + f Jyw + g Jzw - h Jxyzw) , 



* It is interesting to note the mode in which the more general relation connecting cos ( a + + 7), cos a, cos/3, cos 7, 

 >asses over into this on putting 7 = in the former. The result of the substitution is 



COSo 1 COs(cc + /8) COS |8 + 2c0SaC0s(a + j8) 



cos/3 cos (0 + (8) 1 cos a + 2cos/8cos(a + j8) 



1 coso cos/8 cos (a + /8) + 2 cos a cos /3 



COS (a + 0) COS/8 COS a l + 2c0SaC0S/8c0s(a + /3) 



vhere the elements of the 4th column are easily transformed into zeros with the exception of the last element which 

 lecomes 



1 + 2cosocosj8cos(a + /8) - cos 2 (« + /8) - cos 2 /8 - cos 2 a, 

 that the value of the determinant is seen to be 



COSo 1 cos(a + /8) 



cos/8 cos(o + /3) 1 



1 cos o cos /3 



ith this mode of degeneration may be compared that seen on p. 377 of Proc. Roy. Soc. Edin., xx 



