216 DR THOMAS MUIR ON 



according as m is even or odd : and therefore the aggregate of the last m terms is 



- K + (a„„, . . . , a m+l ) (a„„ . . . , a 2 ) or - K + (a 2m , . . . , a m+2 ) {a m _ v ...,%) 



according as m is even or odd : from which we see that in either case the gross aggregate 

 is 0, as was to be proved. 



Instead of using the cofactor notation ' cof ' we might have taken a hint from a usage 

 in the exposition of the theory of general determinants, viz., where the cofactor of a rs in 

 j Oi„ | is denoted by A rs . Our theorem would then stand thus : — 



If&i, a 2 , . . . , a n be the elements of a simple continuant K, and A l9 A 2 , . . . , A n their 

 respective cojactors, then 



a^ - a. 2 A 2 + a 3 A 3 - . . . = K or 



according as n is odd or even. 



Without any notation for cofactors at all, it may however be neatly written as follows : — 



«1 («2» «3> • • • > a n) 

 ~ «2 ( a 3» • • • . a ») a l 



+ a. A (a 4 , . . . , a n ) (a v a. 2 ) 

 - a 4 (o 6> . . . , a„) (a v a 2 , a 3 ) 



( - ^-^(a^gOg a n _ x ) = (a v a. 2 , . . . , a H ) or , 



the seemingly cyclical permutation of the elements arising from the act of reversing the 

 order of the two parts of each cofactor. 



It is when stated in this last form that its suitableness for proving the theorem of 

 § 13 is most readily recognised. Thus, to take the second instance there given, we have 



(a, b) (c,d,e,f) - abc(d,e,f) + a(b,c)d(e,f) - a(b,c,d)ef + a(b,c,d,e)f 

 = (c,d,e,f) - a{ b(c,d,e,f) 



- c(d,e,f)b 

 + d(e,f)(b,c) 

 -e(f)(b,c,d) 

 +f(b,e,d,e) } , 

 = (?,d,e,f) + a(b,c,d,e,f) , 

 = (a,b,c,d,e,f) . 



(16) The next point to be noticed is the effect of changing (a,b) into ab, (b,c) into 

 be, or (c,d) into cd in the identity 



| (a,b) ac ad = (a,b,c,d). 

 (b,c) bd 

 (c,d) 



Such a change is clearly equivalent to subtracting (c,d), ad, or (a,b), — and therefore the 

 right-hand side must be simultaneously changed into 



a(b,c,d), 

 (a,b)(c,d), 

 or (a,b,c)d . 



