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



diately produced on the application of a gentle heat, whereas in 

 the latter only very slowly when the mixture is left some days at 

 the ordinary temperature ; and though a minute trace of nitro- 

 prusside may be formed in this latter case on heating such a 

 mixture, it is only, as I stated before, when the chlorate is in 

 great excess, and therefore enables us to determine the pre- 

 sence or not of that salt by other means before applying the test 

 for nitric acid. 



As to the case of hypochlorous acid, that acid also requires to 

 be in considerable excess in order to form a nitroprusside ; and 

 as hypochlorous acid is endowed with such marked chemical 

 properties, there is little fear of any mistake occurring from that 

 source. 



Finally, in the greater number of cases where it is desirable 

 to determine the existence or not of nitric acid, those compounds 

 of chlorine and oxygen are not at all likely to occur. 



III. Note on a remarkable Modification of Sturm's Theorem, 

 and on a New Rule for finding Superior and Inferior Limits to 

 the Roots of an Equation, By J. J. Sylvester, F.R.S.* 



I 



N my paper on this subject in the preceding Number of the 



Magazine, I showed how by means of the quotients 



fx 



a Y x + b x ; a$x + b% ; . . . a n x + b ni obtained by throwing J -j- under 



jx 



the form of a continued fraction, the process for finding the sig- 

 naletic index for any given value of x in the series for determi- 

 ning the number of real roots of fx within given limits was 

 reduced to performing two sets of n multiplications and as many 

 additions or subtractions. But by means of a very simple obser- 

 vation, I can now show that the second and more laborious set 

 of multiplications may be dispensed with and replaced by the 

 simple operation of finding reciprocals, which can be done by 

 mere inspection by means of Barlow's or similar tables, which 

 are familiar to all computers. If we call the quotients 



a^ + 6,; agx + bt', . . . a n x + b n , 



we must, as explained in the preceding article, find the (n) nu- 

 merical values fjL x , Hey . . . fi n which these quotients assume for 

 any assigned value of x. This being done, the signaletic index 

 corresponding to such value of x 3 i. e. the number of continua- 

 tions of sign iu the signaletic series 



1> 9*» PifHt~ l > MSM2M1-M3-M1, &c., 

 * Communicated by the Author. 



