COAXIAL MINORS OF A DETERMINANT OF THE FOURTH ORDER. 337 



Similarly we have for the equation 



cos _1 a; + cos'V + cos -1 ? = 



the purely algebraical equivalent 



and for the equation 



1 X z 



x 1 y 



z y 1 



= 



COS -1 £ + COS -1 ?/ -+- C0S _1 2 + COS 7^0 + cos _1 v = 



a purely algebraical equivalent essentially the same as that referred to in section 1 5 as 

 giving the relation between cos (a + ft + y + S) and cos a, cos ft, cos y, cos 8. 



17. This suggests a very simple and perfectly symmetrical mode of expressing the 

 relation of section 6 between the coaxial minors of an order lower than the fourth, viz.: — 



G 



COS" 



2J — B 1 B 2 B i 



+ COS" 



a 



2J — B 1 B 3 B a 



+ cos -1 



+ 



C 4 



2V-B 2 B 3 B 6 2J--B&B 



= 



The law of formation of the denominators is perhaps not clear, but this is due merely 

 to a defect in the notation. If we substitute for B's and C's their values as given in 

 terms of the coaxial minors of I a^c^ I we have 



2 



COS" 



I «A C 3 I 



a i I ^2 C 3 i ~~ ^2 I a i C 3 I — C 3 I a i^2 I "+" 2«!&2 C 3 



= 0; 



2(-l)»(|a 1 6 2 | - a!& 2 )*( I a^Cg I - a x c 3 )\\\c z \ - 5 2 c 3 )' 

 md, further, if we denote by 



000 

 he determinant got from | aj).^ | by changing the elements of the primary diagonal 

 nto zeros, the relation may be written 



2 



COS" 



000 



9, - 



a x b 2 



a i c i 



00 ! 00 



&2 C 3 



00 



r 



= 0. 



18. Another matter which has light thrown upon it by certain of the preceding 

 iaragraphs is Sylvester's original illustration of the dialytic method of elimination as 

 pplied to ternary quadrics. It will be remembered that from the equations 



Bx 2 -2G'xy + Ay 2 = 

 Cy 2 -2A'yz + Bz 2 = 

 Az 2 -2B'zx+Cx 2 = 



e deduced three others 



C'z 2 + Cxy-A'zx-B'yz = 

 A'x 2 + Ayz — B'xy — C'zx = 

 By + Bzx - C'yz - A'xy = 



