45IE Mr. J. J. Sylvester on a remarkable Modification 



tions whatever. This is what I term the theory of interpositions, 

 upon which I do not propose here to enter, but which will be 

 fully developed in a memoir nearly completed, and which I 

 shortly propose to present to the Royal Society, wherein will be 

 found combined and flo\ving into one current various streams of 

 thought bearing upon this subject which had previously existed 

 disunited, and appearing to follow each a scpai'ate course. 



7 New Square, Lincoln's Inn, 

 May 13, 1863. 



Remark, 



I am not aware that anyone has observed what the effect would 

 be of omitting to change the signs of the successive residues in 

 the application of Sturm's method, i. e. of employing a proper 



in lieu of an improper continued fraction to express jrij s-sn^fii 



Although easily made out, it is well worthy of being rem^rteiS? 



Suppose 



^ I aouJY-f lounu r-rxav) m'ioai{^ 



-7i = Q- 1 'Uq baauaxo sd oi aqoff I o« 

 \j([^^^ \Xi Lim^i ' iAi n.^V-H*. >*-*.> ja0oiiin>jjtiiii ddi lo aao oJ §ni 

 alo^g ^(bn'jfiB x,id:^il fc /in: t? j/ir*?L- iU' mss' haa Id-gmsau iBdi 



and in general (P being any letter) use P to denote —P. No# 

 we may write 



P1 — Q3P2 — P3 



n —O n —n ^^fH 'io ttttl 



P3=Q5p4-p5 .ndfw 



P4=Q6pa'-P6' 



&C. = &C. <' 



This gives 



f=Q:i<t+P± 



^=^Pl+pQ 



Pl = Q^Pi + P3 



P2 = Q4p3 + P4 



P3=Q'5P4-^P5 

 &C. = &C. 



The law evidently being that the quotients change their sign 

 alternately, i. e. in the 2nd, 4th, 6th, &c. places, and remain 



