Page:Philosophical magazine 21 series 4.djvu/372

348 The facts of electro-magnetism are so complicated and various, that the explanation of any number of them by several different hypotheses must be interesting, not only to physicists, but to all who desire to understand how much evidence the explanation of phenomena lends to the credibility of a theory, or how far we ought to regard a coincidence in the mathematical expression of two sets of phenomena as an indication that these phenomena arc of the same kind. We know that partial coincidences of this kind have been discovered; and the fact that they are only partial is proved by the divergence of the laws of the two sets of phenomena in other respects. We may chance to find, in the higher parts of physics, instances of more complete coincidence, which may require much investigation to detect their ultimate divergence.

Note. — Since the first part of this paper was written, I have seen in Crelle's Journal for 1859, a paper by Prof. Helmholtz on Fluid Motion, in which he has pointed out that the lines of fluid motion are arranged according to the same laws as the lines of magnetic force, the path of an electric current corresponding to a line of axes of those particles of the fluid which are in a state of rotation. This is an additional instance of a physical analogy, the investigation of which may illustrate both electro-magnetism and hydrodynamics. 

LII. Remarks on Mr. Cayley's Note. By.

DESIGNATING by u, v two rational u-valued homogeneous functions of the roots of the equation

$x^{m}=A_{1}x^{m-1}+A_{2}x^{m-2}+\dots+A_{0}=0$

we find by Lagrange's theory that

{{MathForm2|(e)|$$\left.\begin{array}{l} v=\mu_{n-1}+\mu_{n-2}u+\mu_{n-3}u^{2}+\dots+\mu_{0}u^{n-1}\\ u=\nu_{n-1}+\nu_{n-2}v+\nu_{n-3}v^{2}+\dots+\nu_{0}v^{n-1}\end{array}\right\} $$}}

in which $$\mu_{n-1},\ \mu_{n-2},\dots\mu_{0},\ \nu_{n-1},\ \nu_{n-2},\dots\nu_{0},$$ are symmetrical functions of the roots of the original equation in x; and u, v depend separately on two equations of the nth degree

$$\alpha_{1},\ \alpha_{2},\dots\alpha_{0},\ \beta_{1},\ \beta_{2},\dots\beta_{0},$$ being, as well as $$\nu_{n-1},\dots\nu_{0}$$, symmetrical functions of the roots of the equation in x.

I ought to observe that any coefficient, $$\mu_{n-s}$$, in the equation