604 
PROFESSOR SYLVESTER ON THE REAL 
(19) The proposition above established leads immediately to the theorem and corollary 
following, viz. 
Theorem. If c 15 c 2 , . . . c n be a series of ascending or descending magnitudes, and m 
any positive integer, the equation 
Xi(#+Ci) m +X 2 (#+c 2 ) m +. • 
cannot have more real roots than there are changes of sign in the sequence X 15 X 2 , ... 
K, (-)%• 
For obviously (1, Cj), (1, c 2 ), ... (1, c n ) will be points corresponding to rays within a 
semirevolution, and therefore forming a type-pencil. 
Corollary. If the above equation be transformed by any real homographic substitu- 
tion into the form 
^i(^+7i) m +/ /, 2(3'+72) m + • • • +^«(y+7») m =0? 
where y 1; y 2 , ... y n are taken in ascending or descending order, the number of changes 
of sign in the series p,, [a 2 , . . . (*„, ( — ) m [M is invariable ( 25 ); for the effect of any such 
formation will be to leave the type-pencil unaltered except in its phase. 
(20) If we look to the undeveloped form of the superlinear function 
S == g X U™ -f- -]-... -J- 
and are supposed to possess no knowledge of the coefficients which enter into the linear 
elements u, we may still draw some general inferences as to the limit of the number of 
real roots in S=0. Thus if the number of positive units g is j, and of the negative 
units k, and j is not greater than k, it is obvious that, whatever may be the form of 
the type-pencil to S, its variation-index cannot be more than 2 j when m is even, nor 
more than 2 j -f-1 when m is odd ; for the arrangement the most favourable to the large- 
ness of the number of the real roots is that where every two rays with the signs belong- 
tlie X, jj quantities being any real quantities whatever. Then I say — 
1. If T is the type-pencil (per-rotatory or trans-rotatory) of any superlinear form E, every derivative of T of 
the contrary name is the type-pencil of some first derivative of E, as shown in art. (18). 
2. A derivative of T of contrary name may he found such that its variation-index shall he less hy a unit 
than that of T itself, as shown in art. (16). 
3. Hence if i is the variation-index of the type-pencil of E, an ith derivative of E may he found such that 
its variation-index shall he zero, and consequently having no real roots. 
Hence, finally, since the number of real roots of any rational integral homogeneous function in x, y cannot 
exceed by more than i the number of the real roots in any of its ith derivatives, F cannot have more real roots 
than there are units in the variation-index of its type-pencil. 
The subtle point of the argument, it will be noticed, lies in forming the conception of the variation-index to 
a trans-rotatory pencil, in which the singular phenomenon occurs of a reversal of relative polarity in passing 
from the last ray to the first, whereas in a per-rotatory pencil any ray indifferently may be regarded as the 
initial ray, no such reversal in that case taking place. 
( 25 ) It may be noticed that, contrariwise, the limit to the number of real roots given by Newtox’s criteria 
is not an invariant ; it fluctuates with the homographic transformations operated upon the equation ; and a 
question suggests itself as to the maximum value the number of imaginaries indicated by the rule can attain. I 
presume this maximum is not in all cases necessarily the actual number of the imaginary roots possessed by 
the equation. 
