488 Ma. SYLVESTER ON THE THEORY OF THE INTERCALATIONS 
irreaterthan the latter, the effective scale of interpositions will begin with a root of/r ; 
if it be less, the scale will begin with a root of If instead of beginning with +=c 
and ending with -c» we begin and end with any two limits, a and b respectively 
(making abstraction of all roots of/r or of fx lying outside these limits, and forming 
the effective intercalation scale with the roots comprised within these limits exclusively), 
we shall obviously obtain a similar result, but with the condition that the changes rom 
+ to - will be in excess if an even number of A’s and ,’s combined be cut off by 
the superior limit, and the effective scale begin with an h, or if an odd number of h s 
and ,’s combined be so cut off and the scale begin with an , ; and in defect if an odd 
number of A’s and ,’s combined be so cut off and the scale begin with an h or an even 
number be so cut off and the scale begin with an If, now, suppos.ng/r to be of «, 
and M of not more than n, say («) dimensions, we form the signufefic senes /a, 
B„ ...B„ (where the B„ B„ ...B. are the Bezoiitian secondaries or simplified suc- 
cessive residues corresponding to | expanded under the form of an improper con- 
tinued fraction), it may be shown, in the same way as for Sturm’s theorem, that 
whenever^ changes from + to - a change of sign will be gained in the series, and 
when from - to -f a change will be lost ; and that no change can be gained or lost 
except as x passes through the successive real roots of/r. Hence the difference 
between the number of changes of sign in the above signaletic senes when x is taken 
(a) and the number of the same when x is taken (b), will indicate the number o 
roo’ts of /x remaining in the effective scale of interpositions formed between such of 
the roots of/x and of fx as lie between (a) and (h ) ; calling the one nuinber I(« ano 
the other I(i), the sign of I(i) - 1(a) depends not on the relative magnitudes of (a and 
(b) but upon the manner in which the effective scale commences; if I(n)-l(o i- 
positive, the effective scale formed between the (a) and {b) will commence with a 
root of /x; if negative, it will commence with a root of ?i(x). 
Art. (49.). In forming the scale of effective interpositions, it is evidently not neces- 
sary to go on reducing the (h) series and the , series separately and alternately ; the 
same result will be effected more expeditiously by eliding simultaneously any even 
nuinber of /t’s that come together without being separated by an ,, and an; e\en 
number of ,’s that come together without being separated by an (A), and, repeatin 
this process of simultaneous elision, as often as may be required, until no turn /i 
or ,’s come together. Thus, for instance, denoting the magnitudes of the senes o i ta 
mots of f and of ® by the distances of h and , points taken along a riglit line tom a 
;lfnt therein, and supposing such series of roots between the limits a and A to he 
hhhnnnhnnhn'iinhhnhnl^ 
oui- first reduction brings this scale to the form 
hnhhnnhnhh-, 
the next reduction brings it to the foiin 
hnnnli n 
a," Cfq 
