The Collected Mathematical Papers of James Joseph Sylvester volume 4

Cover The Collected Mathematical Papers of James Joseph Sylvester volume 4
The Collected Mathematical Papers of James Joseph Sylvester volume 4
James Joseph Sylvester
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Alone. Then D will be a function of the same type as F(a, a lt a 2, ... ).
Suppose that AZ> = ; that is, that the variation of D due to the change of x into x + ey vanishes in virtue of the equation F^ = 0.
Let D become D when y receives an arbitrary variation y + rju, where 77 is an infinitesimal constant and u an arbitrary function of x ; then the varia tion of D will vanish when x is changed into x + ey + e n u, and consequently when x is changed into x + ey the variation of D will also vani
...sh. Hence and if we take the difference of the variations of D and D, we shall find Now, the arbitrary nature of the function u shows that we must have 352 Lectures on the Theory of Reciprocants [42 and if we reason on -= D, -y D, ... In the same way as we have on D, we da da^ see that the variation A of each of the second differential derivatives of D will also vanish ; and, pursuing the same argument further, it will be evident that the A of any derivative of D, of any order whatever, with respect to a, ai, 2, ...

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