324 



DR THOMAS MUIR ON THE 



devcrtebrated coaxial minor in terms of the vertebrate coaxial minors and the primary 

 diagonal elements. The general theorem thus obtained is 



h h 



Cl c. 2 . c, = |aM<V---i - S»il»M---l + 2«AI<¥**"--I + 



(A) 



d x d 2 d 3 



It may be viewed as a sort of converse of Cayley's, which in outward form it very 

 closely resembles. 



3. The truth of it may be established by proceeding in the manner just indicated ; 

 but there is another available process which has the advantage of presenting it merely 

 as the ultimate case of a more general theorem, viz., a theorem for similarly expanding 

 a determinant which is only partially devertebrated. 



Taking determinants of the 3rd order, we have in succession and without any 

 difficulty of verification, 



• 



H 



a 3 







\ 



\ 



h 



= |«A c sl ■ 



- aj & 2 C 3 1 > 



H 



C 2 



H 







a 2 a 3 

 \ • h 



Cy Cg Cg 



\ 



*A c sl ~ a \W c ^ ~ h\ a i c z\ + a A c *> 



= |«A c sl - «iA c 3 l - K\ a i c s\ - c sl«AI + 2aAv 



Proceeding to the 4th order, we have with equal simplicity in the first case 



\a 1 l 2 c 3 d i \ - aJfrjCgdJ. ( A i) 



For the next case we have similarly 



(&o ^3 ^4 



h \ h h 



c l C 2 C 3 C i 



d x d 2 d 3 d i 



h 



& 3 l i 



d x d 2 d 3 rf 4 



. a 2 a 3 a 4 

 h \ h h 



- &, 



d x d 3 d t 



d x d 2 d 3 d i 



and as each of the determinants on the right has already been expanded in the ne< 

 form, there is at once obtained by substitution 



