446 Mr. J. J. Sylvester on a remarkable Modification 



of exhibiting an instantaneous changing of relief into depression, 

 or the reverse, simply by the rapid withdrawal or insertion of the 

 mirrors a, b. 



I am, Gentlemen, 



Your obedient Servant, 

 6 Pitt Street, Edinburgh, Walter Hardie. 



March 7, 1853. 



The mirrors e and / have a movement on hinges at h, by 

 which the instrument can be adjusted for various distances. 



LXXI. On a remarkable Modification of Sturm's Theorem. 

 By J. J. Sylvester, F,R.S,* 



LET me be allowed to use the term improper continued frac- 

 tion to denote a fraction differing from an ordinary con- 

 tinued fraction, in the sole circumstance of the numerators being 

 all negative units instead of positive units, as thus : 



-&C. 



The successive convergents of such a fraction as that written 

 above will be 



If we call these respectively 



we have the general scale of formation 



Moreover, we shall have universally 



N^ . Dt_, — Nt_i . D^ equal to + 1, 



instead of alternating between + 1 and — 1, as is the case in 

 continued fractions of the ordinary kind. 



Again, let me be allowed to use the term " signaletic '^ series to 

 denote a series of disconnected terms, designed to exhibit a cer- 

 tain succession of algebraical signs + and — , and to speak of 

 * Communicated by the Author. 



