445 
EXPRESSED IN TERMS OF THE ROOTS OF (j)X, fx. 
inquiry into the numerical relations which exist between the entire series of forms 
'^v,i-v R given value of i, corresponding to all values of v between 0 and i inclusive. 
In order to avoid a somewhat oppressive complication of symbols, I shall take 
a particular numerical example, i. e. yn—l w =6 2 = 4 , and compare the values of 
^ 0,4 5 ^ 1,3 j ^ 2 , 2 ? ^3,1 5 ^4,05 ^f which wc know to be identical, [to a numerical 
factor with one another and with the second simplified residue to /and that 
being of the fourth degree in x ; our object in the subjoined investigation is to deter- 
mine the numerical ratios of these several forms of to one another. 
First. Let v=0 i/=4. The leading coefficient is 
' y h \ ^2 ^3 ^4 ^5 ^6 ^7 
^2 n-i yii 
which we know a priori (it should be observed) to be essentially an integral 
function of the h and the n system. In this, the term containing nl will be evidently 
(j^ j 
^5 
J7l ‘< 1-2 % ^4 
the r] system to which the latter summation relates being now reduced to consist of 
’?2 ^3 ^4 ^ 5 - In this expression, again, the coefficient of nl is evidently 1 . Hence, 
therefore, the leading coefficient in ^o .4 contains the term nl-nl- 
Secondly. Let v=l i'=3. The leading coefficient in ^, 3 becomes 
^1 ^2 ^3 
-^1 
X 
''li ^6 
_ A2 A3 A4 A5 Ag hy 
^3 ^4 ^5 Ag hy 
X 
?74 nt ^6 
^2 ^ 3 _ 
In this, the factor affecting will be 
^1 ^2 ^3 
A, 
X 
^4 ^5 
_ A2 A3 A4 Aj Ag Ti 2_ 
A2 A3 A4 A5 / 
ig hy 
X 
n ^ n-o 
^2 ^ 3 _ 
rie being now understood to be eliminated out of the ?] system included within the 
above summation. Again, in this latter sum the factor affecting will be 
(B.) 
^2 ^3 
X 
^4 
A2 A3 A4 Ag Ag hy 
A2 A3 h^ Ag Ag hy 
_A. 
X 
^4 
^2 
fi; and riQ being now both eliminated out of the rj system. This last sum can of course 
only represent a numerical quantity. 
