THE DIFFERENTIATION OF A CONTINUANT. 215 



By way of proof we note that the first of the n— 1 terms on the left is 



(«1> a i) («3» a 4> • • • > «») » 



and that therefore we may write instead of it, 



K — a^a^ . . . , a„). 



Then, as the second of the n — 1 terms is 



- a x a 2 a 3 {a v ..., a n ) 

 the aggregate of the first two is 



K - a x (a 2 , a 3 ) (a 4 , . . . , a„) . 



Similarly, the third term having the factor (a 2 , a 3 ) in common with this aggregate, the 

 aggregate of the first three is found to be 



K - a x (a 2 , a 3 ) (a 6 , . . . , a n ) . 



This process being continued it is seen that the aggregates of odd numbers of terms- 

 follow one law, and the aggregates of even numbers another law, viz., 



Terms. Aggregate. »£*_ Aggregate. 



I K - a x (a it .... a n ) , 2 K - a^(a 21 a 3 ) (a 4 , . . . , a„) , 



3 K - a t (a 2 , a 3 ) (o 6 , . . . , a„) , 4 K - a^, ...,a 5 ) (a 6 , . . . , a„) , 



5 K - a^, . . . , a 5 ) (a 8 , . . . , a n ) , 6 K - aj(a 2 , . . . , a 7 ) (a s , . . . , a„) , 



From the first column we learn that when n is odd the aggregate of the n—X 



terms is 



K - ^(og, ...,«„) 

 i.e., (a 3) . . . , a„) , 

 as was to be proved. 



Also, from the second column it is seen that when n is even the aggregate of n — 2 



terms (i.e., all the terms except the last) is 



K - a u (a,, . . . , a„_i) a n 



to which if we add the last, viz., + a x (a 2 , . . . , a n _^) a n , we obtain the aggregate 



K 

 as desired. 



(15) There is however a more elementary theorem which leads easily up to this, 

 and which can be more readily proved in the same way. It has to be noted too, that 

 from some points of view it is of greater interest than that to which it leads. It is : — 



If 'cof a r stand for the cofactor ofa r in the continuant K(a,, a 2 , . . . , a n ), then 

 a l cof flj - a. 2 cof a 2 + a 3 cof a 3 - . . . = K or 

 according as n is odd or even. i 



To condense the proof we may take advantage of the fact that (oj, ...,«„) = 

 (a n , . . . , a x ), and sum the half of the terms from the beginning and the half from the 

 end. Thus, when n is even, equal to 2m, the aggregate of the first m terms is found 

 to be 



K - (oj, . . . , a m ) (a m+l , . . . , a 2m ) or K - (a 1} . . . , a m _^) (a m+ .,, . . . , a 2m ) , 



