THE DIFFERENTIATION OF A CONTINUANT. 



219> 



let us first perform on it the operations 





(row)! - *(row) 2 , 

 (row) 2 _ -(row),, 



(row) 5 - - (row) fi , 



and then, to preserve the skewness, the corresponding operations 



(COI)! - -(C0l) 2 , 



(col) 2 - - (col) 3 , 



(col) 5 - - (col) 6 . 

 The result of this is a zero-axial skew determinant which is the square of 



{a,b,c) 



a 







c 



b 





' 





(b,c,d) 



b 







d 



c 









(c,d,e) 



c 







e 



d 

 (d,e,f) 



f 



- £ 



e 

 1 



Developing the Pfaffian in terms of the elements of the first frame-line and their 

 complementary minors, we see that it 



_ (a,b,c) j (c,d,e) , ,, c I a b (e,f) 

 ~ ~^~{ e " KJ) el b"c' T 



= (3M.( Ci d A f) - a -(e,f) 



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



as was to be proved. 



(23) If, on the other hand, we begin with (a, b, c, d, e,f) 2 we must try to trans- 

 form its equal factors in such a way that the subsequent application of the multiplica- 

 tion-theorem may produce the left-hand member. 



"Writing the factors in the form 



a 



1 













1 



- a 









. 



-1 



b 

 -1 



1 



c 



-1 



1 



d 



-1 



1 



e 

 -1 



1 

 / 



> 



h 

 -1 



1 



1 



d 



-1 



-1 



- c 

 1 



1 

 / 



-1 

 — e 



1 



