Mr. T. Muir on a Theorem in Continuants. 



137 



the eyepiece, or by bringing the rotating wire of the eyepiece 

 into apparent contact with the edge of the face in question. 



The method is available and chiefly useful in cases where it 

 is almost impossible, or at least requires a refined lapidary's 

 skill to form an artificial twin according to the elegant method 

 of Des Cloizeaux. 



X.B. — As a contribution tending to complete my memoir 

 of December 15, I add three more figures representing various 

 forms of the crystals of Ludlamite. 



rig. 3 



Figr.4. 



January 18, 1877. 



XIX. A Theorem in Continuants. 

 By Thomas Muir, M.A., F.R.S.E* 



IX the Philosophical Magazine for June 1853 (vol. v. p. 

 453) Professor Sylvester gave the first imperfect germ of 

 a theory of continuants— determinants of the form 



h 0-0 



a t 



— 1 a 2 h 



-1 a 2 



-1 





 h 



«4 



or K(a 1 5 i a 2 *2 a^ a 4 ) 



— pointing out their connexion with continued fractions, as 

 seen in the identity 



\ h _ K(a^ « 2 *2 ...a n ) m 



a 3 + . 



+ 



b n -l 



K(a 2 6 2 ...a n ) 



and in the next volume (pp. 297-299) he established a funda- 

 * Communicated by the Author. 



