214 DR THOMAS MUIR ON 



and as this has two factors in common with the first term inside the rectangular 

 brackets, the original expression takes the form 



• {(flfc+i, . . . , a n ) (a v . . . , a y _i) + (- lJr-P+ifa, . . ., a p -i) (a y+1 , .... On)} . 

 Making a second use of the above-mentioned general theorem we can substitute 



(a v . . . , a„) (ap+i, . . . , a y _i) 



for the expression inside the double-curved brackets. The original expression is thus 

 resolved into six factors, one of which is K, and the others 



('<!, • • • , Oa-l), (fflo+1, • • • , «0-l), ( a P + h • • • , a y-l)) («y+l. ■ • • , «S-l), («8-»-l, • . . , «n) , 



the product of which, from §§ 4, 5, we know to be 



cof a a apa y a$ . 



(13) The next auxiliary theorem used in § 9 is that which affirms that 



(a,h) (c,d) - abed + a(b,c)d = (a,b,c,d) * 

 and 



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

 or that 



cof ab . cof cd - coiae.coibd + cof ad. cof be = (a,b,c,d) 

 and 



cof ab . cof cdef 



- cof ae . cof bdef 

 + cof ad . cof beef } — (a,b,c,d,e,f) , 



- cof ae . cof bed/ 

 + cof of .coibede 



The general theorem which includes these concerns (a 1} a 2 , . . . , a 2n ), and is 



= (QSj, a 2> . . . , &.,„) j 



cof fljOSg . cof « 3 « 4 a 5 . . . ffljn 

 - cof ajGfg . cof a 2 a A a r) . . . a 2 „ 

 + cof a l a i . cof a 2 a 3 a 6 . . . a. lH 



and there is a corresponding theorem for the case where the number of elements is odd, 

 viz., 



cof a x a 2 . cof a 3 a 4 a 5 . . . a. 2 „ +1 \ 

 - cof a^g . cof a 2 a 4 a 5 . . . a 2re+1 I = (« 3) a 4 , . . . , a 2re+1 ) . 



+ 



I 



(14) The two theorems may therefore be enunciated together thus : — 



If cof x denote the cof actor of x in the continuant K (a u a 2 ,..., a n ), then 



2,(- 1) T cof a x a r . cof a x a 2 . . . aja x a r = K , 



or = cof a x a 2 , 

 ■according as n is even or odd. 



* It should be noted that this is also a case of theirs* auxiliary theorem. 



