then 
and also 
460 MR. SYLVESTER ON FORMULA RELATING TO STURM’S THEOREM. 
Let us commence with the case where z=0, we have then 
we have thus 
[It may easily be verified that the negative sign interposed between the two parts of 
the right-hand member of the equation has been correctly taken, for 
) contains a term 
coutaios u term .h^ 
2{i„ K-h,J contains a term -li’.-.-k'.-,, 
and thus the term which does not contain k,h...k^, will (as 
it ought to do) disappear from the right-hand side of the equation.] 
Now suppose 
ki — /r2) 
^{k,h...kj=0, 
except when one or the other of the two disjunctive equations 
qi, 92, g3---9m-i=l5 3, 4..,m 
qi, 92 , ?3--gm-i=25 Sj 4...W 
is satisfied (by a disjunctive equation, meaning an equation which affirms the equalit\ 
of one set of quantities with another set the same in number, each with each, but in 
some unassigned order), 
tlcncc 
=:2kj ks ...kj^iki ks ...^m)- 
Hence when 
k, = {-~Yh ’■m -2 becomes Y^{kg^kg^..\^_,)^{h h...k^), 
i. e. 2Uk. h...k,„){k,^h,K^...K^_,-^^hh...kJ, 
the S referring to supposed to be disjunctively equal to 3, 4, ....m. 
Now r ^_2 is of {ni—2) dimensions in x, and whenever more than one equa ity 
exists between the k\ \ and both vanish (in fact every term in each vanishes 
separately), and therefore t„_„ which 
Hence {—T‘r ^-2 must be always of the form 
K-K.-, i KJ’ 
