COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 



333 



This, however, is well known to be equal to the determinant 



x h Jbc h Jca I Jab 



h Jbc x I Jab k Jca 



k Jca I Jab x h Jbc 



I Jab k Jca h Jbc x 



and we may consequently say that the rationalizant of the expression x + hjbc 

 + kjca + ljab is the biaxisymmetric determinant of the 4th order which has the 

 terms of the expression for the elements of its first row, all the elements of its 'primary 

 diagonal alike, and all the elements of its secondary diagonal alike. 



Another determinant form of the result is obtained by using the dialytic method of 

 elimination. Taking the original equation and multiplying in succession by Jbc, 

 Jca, Jab, we have 



x + h Jbc + k Jca + I Jab = 



hbc + x Jbc + lb Jm -f kc Jab = 



kca + la Jbc + x Jca + he Jab = 



lab + ka Jbc + hb Jca + x Jab = 



and therefore on eliminating Jbc, Jca, Jab there results the rationalizant 



X 



h 



k 



I 



hbc 



X 



lb 



kc 



kca 



la 



X 



he 



lab 



ka 



hb 



X 



It is easy to change the one form into the other ; indeed, this change is what has been 

 effected in sections 8, 9, Professor Nanson having obtained his results in the latter 

 )f the two forms. 



13. Another closely related problem, as Professor Nanson has made clear, is that of 

 xpressing cos (a + /3 + y) in terms of cos a, cos ft, cos y, or say, for shortness' sake, S in 

 ;erms of A, B, C. 



Since 



cos(a + /8+y) — cos a cos cos y + cos a sin /3 sin y + cos /3 sin y sin a + cos y sin a sin fi = , 



ve have 



S - ABC + AJ1-B 2 J1-C 2 + BVl-Cyi-A 2 + C^l-Ayi-B 2 = 0, 



tnd the problem is seen to be a case of the preceding, the result being either 



S-ABC AVUByi^C 2 BJUTCZJI^A 2 C Jl^A 2 Jl^B 2 



AJl^B2jl^Q2 S-ABC CjT=A 2 Jl=B 2 B Jl^C 2 Jl=A 2 



Bjl-CyT3A2 CjT=A?Jl=W S-ABC A Jl^B 2 Jl^C 2 



Vjl-A 2 J1^B 2 BJ1^C 2 J1^A 2 AJU-B 2 jT^C 2 S-ABC 



