MR. SYLVESTER ON FORMULA RELATING TO STURM’S THEOREM. 
461 
denoting some integral function of (m~2) dimensions in respect of the system of 
quantities The result above obtained enables us to assign the value of 
k^) 
when = 
viz. k,^k,^ k,^...kr^__,)-{-^h.k,...k,^. 
Now for a moment suppose, selecting (m-—!) terms k^, k^, k^...k^ out of the m terms 
of the k series, that 
Cl{k, k^k^...k^, k^)~k^-^—k^-\S,()^, 
Jz ^c. -|-^2 ^m — siki k^. . .k^, 
where Sj means that the quantities which it governs are to be simply added together, 
S2 denotes that their binary, S3 that their ternary, and in general S^ that their r-ary 
products are to be added together. 
When k^=k2, CL becomes 
k^-^—k^-\k,+^,{k^k,...k^))-\-k'^-\[k^ S,(k^k^...k^)-C^^{hh...kJ) 
— k^-\ [k, S2(/f3 A’4 . . . k ^) + 83(^3 A-4 .. . Ar j) +&c. + A. (A, S^_4(A3 A4 . . . A^) + S^_3( A3 k,... kj) 
±2S„_2(A3 A4...AJ, 
which evidently equals 
+ {2S^_2(A3 A4....AJ + A, S«_3(A3 A4...AJ}, 
i. e. +{Ai 2(A,^ A,^...A,^_,)+2A3 A4...A„}. 
Hence when ki=k^, T=n, and 
xn(A„ A,^...A,^_,AJ ; 
and so in like manner, when A, is equal to any one of the (m — 1 ) quantities k^, k^...k^, 
the form of r„_2 above written will have been correctly assumed. But r^_2 may be 
treated as a function of (m — 2) dimensions in A„ and consequently any form of 
{m — 2 ) dimensions in A,, which fits it for (w— 1) different values of A,, must be its 
general form, and accordingly we have universally, 
Art. ( 39 .). With a view to better paving our way to the general form of r for all 
values of i, let us pass over the case of i=\ and go at once to the equation 
and to better fix our ideas let m—J, so that the equation becomes 
t^.fx—7^.fx-\-'^^=0 ; 
we have then, preserving the same relation as before [i. e. using h to denote any root 
