550 Royal Society. 
Norway, published a remarkable theorem, which gives the sum of a 
series of integrals of a more general form, and extending to higher 
powers than those in Euler’s theorem ; and furnishes a multitude of 
solutions for each particular case of the problem. Legendre, though 
at an advanced age, devoted a large portion of time to the verifica- 
tion of this important theorem, the truth of which he established upon 
the basis of the most rigorous demonstration. M. Poisson has, in a 
recent memoir, considered various forms of integrals which are not 
comprehended in Abel’s formula. 
The problem, to the solution of which the author has devoted the 
present paper, is of a more general nature than that of Abel. The 
integrals, to which the theorem of the latter refers, are those com- 
> 
g Pd : 
prised in the general expression ar where P and Rare entire po- 
lynomials in z. Next in order of succession to these, there naturally 
presents itself the class of integrals whose general expression is 
Ff ae where the polynomial R is affected with a cubic, instead 
of a quadratic radical; but Abel’s theorem has no reference to these, 
and consequently affords no assistance in their solution. The same 
may be said of every succeeding class of integrals affected with roots 
of higher powers. Still less does the theorem enzble us to find the 
sum of such integrals as /¢ (R) dz; R being, as before, any entire 
polynomial (that is, containing at least two different powers of 2), 
and ¢ being any function whatever. The author then details the 
processes by which he arrives at the solution of this latter problem. 
March 17.—A paper was read, “On the reciprocal attractions of 
positive and negative electric Currents, whereby the motion of each is 
alternately accelerated and retarded.” By P. Cunningham, Esq., 
Surgeon R.N. Communicated by Alexander Copland Hutchison, 
Esq., F.R.S. 
The author found that a square plate of copper, six inches in dia- 
meter, placed vertically in the plane of the magnetic meridian, and 
connected with a voltaic battery by means of wires soldered to the 
middle of two opposite sides of the plate, exhibited magnetic polarities 
On its two surfaces, indicative of the passage of transverse and spiral 
electrical currents, at right angles to the straight line joining the 
ends of the wires. The polarities were of opposite kinds on each 
side of this middle line, in each surface; and were reversed on the 
other surface of the plate. The intensities of these polarities at every 
point of the surface were greatest the greater its distance from the 
middle line, where the plate exhibited no magnetic action. The au- 
thor infers from this and other experiments of a similar kind, that each 
electric current is subject, during its transverse motion, to alterna- 
tions of acceleration and retardation, the positive current on the one 
side of the plate and the negative on the other, by their reciprocal at- 
tractions, progressively accelerating each other’s motions, as they 
approach, in opposite directions, the edge round which they have to 
turn, After turning round the edge their motion will, he conceives, 
