138 



Mr. T. Muir on a Theorem in Continuants. 



mental theorem regarding them. A fall account of their pro- 

 perties (written without a knowledge of Sylvester's early dis- 

 covery) will be found in the 'Proceedings of the Royal Society 

 of Edinburgh' for the Session 1873-74. 



The following new theorem on the subject seems to me in- 

 teresting, both on its own account and because of the exceed- 

 ingly simple way in which it is demonstrated. 



Consider the continuant 



a t + x 



co -f a x x 















-1 



a 2 



co + a^x 















-1 



«3 



co + a s x 















-1 



« 4 



co -f a^x 















-1 



a 5 



Increasing the elements of the first, second, . . . rows by x 

 times the corresponding elements of the second, third, . . . 

 rows, it departs from the continuant form, becoming 



a x co + a-^x + a^x cox-\-a 2 x 

 ■ 1 a 2 — x co + a 2 x + a^x 



— 1 « 3 — x 



-1 











cox + a- 6 x 



co -f a- d x + a A x cox + a A x 



a± — x co + a±x + a 5 x 



1 Clz 



Now, however, diminishing the elements of the second 

 column by x times the corresponding elements of the first 

 column, diminishing the elements of the third column by 

 x times the corresponding elements of the new second column, 

 and so on, we have 



% 



co + a 2 x 











1 



a 2 



co + a^x 











-1 



a 3 



CO + o A x 











-1 



a 4 co + a 5 x 















— 1 a 5 + x 



which again is a continuant ; hence the theorem 



K(a l + aP +a i*a a »+«** . . . a n ) = K(a 1 (a+a ^a 2 ^+ a ^ . . . a n + x). 

 Glasgow, January 20, 1877. 



