456 MR. SYLVESTER ON FORMUL.® RELATING TO STURMS THEOREM. 
thus t....?.- -r,_=./,+a.=0), when and/, are taken as the primitives, the 
corresponding equation will be of the form 
these two equations must therefore be identical, and consequently (to a 
numerical factor pr^s), so that and 9..-1 are reciprocal forms; this is also obvuous 
from the consideration that must, by the general law of reciprocity (established 
above), be a residue to (/„, <prn-i'), which the latter function itself may be considered 
to be. Or the same thing is obvious directly, by writing 
and then making 
, I, \ ( nn h \ 
where 
A=(~)”-^ {h~h^y . {h—hY ...{h—Kf 
X(A2“^3)"”-(^2~0' 
X (hm—l 
D (D being the Discriminant, more commonly called the 
Determinant to f) ; or finally, 
as was to be shown. 
Section III. 
On the application of the Theorems in the preceding Section to the expression in terms 
of the roots of any primitive function of Sturm’s auxiliary functions, and the other 
functions which connect these with the primitive function and its first dijferential 
derivative. 
Art. ( 3 .''u). The formula in the preceding Section had reference to the case of two 
absolutely independent functions and their respective systems of roots : when tlie 
functions become so related that the roots of the one system become explicitly or 
implicitly functions of the roots of the other system, the formulae will become 
expressible in terms of these latter alone, and in some cases the terms (of which the 
sum is always essentially integral) will become separately and individually represent- 
able under an integral form. Such, as I shall proceed to show, is the case for tvo 
functions, of which one is the differential derivative of the other. When/and are 
