32 Transactions of the South African Philosophical Society. 



The attempt to establish this identity by transforming the one 

 continuant into the other has led to the following theorem, which,, 

 besides effecting a still wider generalisation than that of § 2, places, 

 the whole matter on a different and far more interesting footing. 



5. The value of the continuant 



a b . 



(1 -ri)c a-b - c 2b 



. ' (2-n)c a -2b -2c ...... 



a-(n-2) (b + c) (n-l)b 



- c a- (n - 1) (b + c) 



given in § 2 is not altered by adding to its matrix the matrix of the 

 continuant 



(n-l)x x . 



(l-n)x (n-3)x 2x 



(2-n)x {n - 5)x 



- (n - 3)x (n - l)x 

 -x - (n - l)x 



The result of performing on the latter matrix the first of the 

 operations performed in § 2 on the original matrix, viz., the operation 



row t + row 2 + row, + 



is to make a row of zeros : it follows, therefore, that this operation 

 performed on the matrix, which is the sum of these two matrices, 

 will lead to exactly the same result as in § 2. The same is the 

 effect of the operation 



roWj + 2 row 2 + 3 row. + 



performed on the determinant of order n — 1, which appears after 

 the removal of the factor a - (n - l)b ; and so at every successive 

 stage of the factorization. 



6. As an illustration of this theorem let us take the case of the 

 fourth order, viz., 



a-{-3x b-\-x 



-3c-3x a-b - c-\-x 2b-\-2x 



- 2c - 2x a -2b -2c - x 3b -f Sx 



- c -.x a - e 3b - 3c - 3x 



