THE DIFFERENTIATION OF A CONTINUANT. 213 



the semi-Hessian of K will be 



It is clear however that before we can test this the auxiliary theorems which we have 

 used in proving the first two cases must be generalised. 



(11) The first of these auxiliary theorems is that which involves the fact that all 

 our minor Pfaffians of the second order contain K as a factor. What the theorem itself 

 actually is becomes more apparent if we adopt the 'cofactor' notation instead of the 

 notation of differential-quotients : for then the instances are — 



cof cd cof ce cof cf 



cof de cof df 



cof ef 



cof bd cof be cof bf 



cof de cof df 



cof ef 



= K-(a,b) 



= K- ac , 



Or 



cofcd.coief - coice.coidf + cof cf . cof de = ~K- (a,b) , 

 cof bd . cof ef - cof be . cof df + cof bf . cof de = K- ac , 



(12) The general theorem here exemplified is — 



If K be a simple continuant of any number of elements a 1? a 2 , . . . , a n , and a, /3, y, S 

 be the suffixes of any four elements taken in the order in which they occur, then 



cof a a a p . cof a y a$ - cof a a a y . cof a p as + cof a a a$ . cof a p a y 



= K • cof a a a p a y as . 



In proof of this we note in the first place that the expression on the left 



= (a v . . . , a a -i) (a a+ i, . . . , a P -\) (a p+1 , . . . , a n ) • (a 1; . . . , a y _i) (a y+ i, . . . , a S -i) (a«+i, ■ ■ ■ , a n ) 

 - (a 1( . . . , a a _i) (a 0+ i, . . . , a y -i) (a y +i, . . . , a n ) ■ (a v . . . , a p .{) (a p+i , .. ., a S -i) (os+i, • ■ . , a n ) 

 + (a lt . . . , a a _i) (a a+1 , . . . , a«_i) (a g+1 , . . . , a») • (a l5 . . . , a^-i) («£+i, • • • , « v -i) (« y +i, • • • > a '0 > 



which, on account of there being two factors common to all the terms, 



= (a v . . . , a a -i) (a i+ i, ...,«„)[ (a a+ i, . . . , a P -i) (a p+ i, ...,««) (a^, . . . , a v _i) (« y +i, • • • , «s-i) 



- (a a+ i, . . . , a Y _i) (fly+i, . . . , a„) (a ls . . . , a P -i) (a p+h . .., a S -i) 

 + (a a+1 , . . . , a s _i) (a x , . . . , fl^-i) (ap+i, . . . , a v _i) (a y+1 , . . . , a«)] . 



Again, the last two terms inside the rectangular brackets here have two factors in 

 common, their aggregate thus being 



= («i, . . . , a^_i) (a y+ i, . . . , an) { - {a a+h . . . , a y _i) (a p+1 , ...,a S -i)+ (a a +i, . - . , a«-i) (a p +i, . . . , « v -i)} , 



which, by an extension of the general theorem used in § 2 



= (a„ . . . , o^.!) (a y+1 , ...,«„)• (_ l)y-P+i (a a+1 , . . . , a^-i) (a y +i, . . . , a S -i); 



