464 
MU. SYLVESTER ON FORMULiE RELATING TO STURMS 'IHEOREM. 
Hence then for the two systems of values of h, viz. 
hi~h^ 
h^=.}h or h^=^hs 
h.^hi 
the form of r wiW have been correctly assumed. But since the derived form is a 
linear function of h,, /l, K, this is not enough to identify the assumed with the 
general form, since for such verification four systems of values must be taken, four 
being the number of terms in a function of three variables of the first degree. If, 
however, we had adopted a separation of the multiplicity three into two parts, and 
had started with supposing K=h=h h^h, we should have found that r would 
have become , . . , 
h){ 2 K+^h)- 
Moreover, when these equalities subsist, 
/fg ^3 A’a ^3 /fs+^h kz k\ ^g + ^i ka k^ k^-\-k.2 A'3 k^ k^ 
becomes ‘ 2 k\k^^^k\.kl, and the common factor k\.k^ disappears in the course of the 
operations for finding r, and eventually we have to show (in order to support the 
universality of the previously assumed form foi r) that 
becomes — 2??„ — 8^5 vrhen 
and 
which is evidently true, 
following cases, 
and also for the cases 
= = = 
^55 
Hence then r will have been correctly assumed for the 
k, ” k.2 Aj = A^3 
ki — k.2~k^~k^ ; 
^j = A2=A3 and k^—k^ 
k,~k,~ki and A2=A-4 
A’2=/f5— A:3 and k^ — k^ 
^j=rA’2=:A’4 and k^=k.2 
k^ — k^ — k^ and k^^k^ 
k2=k^=ki and A’l^Ag 
/. e. for eight cases in all, whereas four only would have sufficed. Hence, abim- 
dantid demoiistrationisl'’ the form assumed for r, is in the case before us the geneial 
• form. . e 
Art. ( 41 .). We may now easily wn-ite down the general form which r assumes tor 
all values of i and prove its correctness. If the roots be /q A2 A3---A,,,, and 
