348 Mr. G. B. Jerrard’s Remarks on Mr, Cayley’s Note. 
The facts of electro-magnctism 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 ali 
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 comeidence in the mathematical expression of 
two sets of phenomena as an indication that these phenomena 
are 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 My. Cayley’s Note. By G. B. Jerranp*. 
Lye by wu, v two rational x-valued homogeneous 
functions of the roots of the equation 
a + Ava™—14 Aga? ,.-+-A,=0, 
we find by Lagrange’s theory that 
V=Mn—1 t+ Mn—2U+ fra U +e. ss : 
U= Van FV V+ Va_s¥?e +. ty, otf? (e) 
in which fyi, Mn—2) ++ Mor Vn—1) Yn—2)++Vq are symmetrical fune- 
tions of the roots of the original equation in z; and u, v depend 
separately on two equations of the nth degree 
UP a, UP ont At. eee O, ln) Se 
v+B, y®—14. Bo yn-2 4 Ae +B,=0, oar (V) 
Gh}, tg) + «Any By, Bos,»+ Bn being, as well as y»_1,..¥g, symmetrical 
functions of the roots of the equation in 2. 
I ought to observe that any coefficient, u,_,, in the equation 
* Communicated by the Author. 
