THE DIFFERENTIATION OF A CONTINUANT. 



217 



(17) We are now prepared to show that 



I (a, b) ac ad ae af = (a,b,c,d,e,f) , 



(b,c) bd be bf 



(e,d) ce cf 



(d,e) df 



M 



For, expanding the Pfaffian in terms of the elements of the first frame-] ine and their 

 complementary minors, we have 



(a, b) | (c,d) ce cf 

 (d,e) df 



fe/) 



- ac I bd, be, bf 



(d,e) df 

 (e,f) 



- ae I (b,c) bd bf 



(c,d) cf 

 df 



+ ad I (b,c) be bf 

 I ce cf 



(e,f) 



+ af I (b,c) bd be 

 (c,d) ce 

 (d,e) 



which by § 16 is equal to 



(a,b) (c,d,e,f) - ac . b(d,e,f) + ad . (b,c) (e,f) - ae . (b,c,d)f + af(b,c,d,e) , 



and therefore by § 1 4 equal to 



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



(18) The effect of changing (a,b) into ab on the left of the preceding identity is 



to subtract 



\(c,d) ce cf i.e., (c,d,e,f) , 

 I (d,e) df 



M 



and as the right-hand side equals 



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

 the result is 



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



The other changes taken in order lead to 



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



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



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



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



(19) In this way we reach the perfectly general theorem ^ 



(a v a 2 ) a x a s a x a 4 



(« 2 , « 3 ) a 2 a 4 • 

 (a,, a 4 ) . 



i In 



a 2 a in 



— (a v a. 2 , . . . . , a^n) , 



( a 2n-l) a 2n) 



with which is connected the series of identities resulting from the change of any one of 



* It is worth noting that the elements of the Pfaffian are the two-line coaxial minors of the equivalent continuant. 

 VOL. XL. PART I. (NO. 11). 2 K 



