Mr. J. J. Sylvester on a remarkable Modification 



the present I shall content myself with showing a posteriori the 

 truth of the theorem for a particular case. Let 7i=3. The two 

 series which are to he proved to be signaletically equivalent may 

 be written 



1, A, BA-l, CBA-C-A 



1, C, BC-1, ABC-A-C. 



Gall these respectively S and (S). In S we may substitute in 

 the third term, in place of BA— I, CA without affecting the 

 signaletic value of the series ; for if the second and fourth terms 

 have different signs, the third term may be taken anything 

 whatever, since the sequence of the second, third, and fourth 

 terms will give one continuation and one change, whatever the 

 middle one may be. Suppose, then, that the second and fourth 

 terms have the same sign, and let 



stHo:5i;.7aoD CBA-C--A=m3. A;^^^^ ^^^ '^^ 

 '^ll^-'A-. C(BA~1) = K + 1)aJ^.._- :_L 11 

 ■ ^•^".•. (BA-l).AC = (m2 + l)At '"^^ ' 



Hence (BA— 1) and AC will have the same sign; hence S is 

 signaletically equivalent to S' where S' denotes the serieii^p'^^^r^ 



1, A, CA, CBA-C-A. 



Now, again, if CA is negative, we may put instead of A any- 

 thing whatever, and therefore, if we please, C, without affecting 

 signaletically the value of S'. But if CA is positive, A and C 

 will have the same sign, and therefore on this supposition also 

 C may be substituted for A. Hence always S' is signaletically 

 equivalent to S", where S" denotes 



1, C, CA, CBA-C-A. 



Again, if C and CBA — C— A have different signs, the value 

 of the intermediate term is immaterial ; but if C and CBA — C — A 

 have the same sign, let . : 



CBA:-tf-A=m2C; ■;- •• .'-^^- 



then A(CB + l) = (H-m2)C, 



and A2(CB-l) = (l+m2)AC; 



and consequently CB — 1 and AC have the same sign. In every 

 case, therefore, S" is signaletically equivalent to 



1, C, CB-1, ACB-A-C; 



i. e. S is signaletically equivalent to S', and therefore to S", and 

 therefore to (S), as was to be proved. 



The application of the foregoing theory to Sturm^s process 

 for finding the number of real roots of an equation is apparent ; 

 for a very little consideration will serve to show, that if we 



