Mr. A. Cayley on a Question in the Theory of Numbers. 539 
izing effects, which, though not large, were easily recognized and 
distinguished from the non-action of the glass. Gold-leaf and 
gold films on glass produced a like effect in a camphine-bath, 
the results being easily distinguished from those of the glass 
and camphine only, in places where the glass had been cleared 
from gold. 
Films of palladium, rhodium, silver, a plate with deposited 
gold particles, and a layer of deflagrated silver particles gave a 
like result, the effect varying in degree. The sulphuret of cop- 
per before spoken of as in contrast with the metals, gave only a 
doubtful result, if any. 
Before concluding, I may briefly describe the following nega- 
tive results with the preparations of gold. I prepared a power- 
ful electro-magnet, sent a polarized ray across the magnetic field, 
parallel to the magnetic axis, and then placed portions of the 
ruby and violet fluids, also of their deposits wet and dry, also 
portions of the gold films, of gold-leaf, the results of deflagrations 
&c., in the course of the ray; but on exciting the magnet could 
not obtain any effect beyond that due to the water or glass, 
which in any case accompanied the substance into the magnetic 
field. In some cases very dense preparations of the ruby and 
blue deposits were employed, the intense electric lamp light being 
required to penetrate them. 
I passed the coloured rays of the solar beam through the 
various gold fluids and films that have been described. For 
this purpose a beam of sunlight entering a dark room through 
an aperture 1th of an inch in width, was sent through two of 
Bontemp’s flint-glass prisms, and its rays either separated, or at 
once thrown on to a pure white screen ; the different objects were 
then interposed in the course of the ray, but I could not per- 
ceive when any portion of a ray passed (and that was generally 
the case) that it differed sensibly in colour or quality from the 
ray passing into the preparation. In like manner, the objects 
were put into the differently coloured rays and observed by the 
reflected light, a lens being sometimes employed to concentrate 
the light; but I could not find any marked difference between 
the colour or character of the ray reflected and the impinging 
ray, except in quantity. 
LX. Solution of a Question in the Theory of Numbers. 
By Anvuur Caytey, Esq.* 
ib is well known that if N=a%b’... where a, b,.. are primes, 
then oN, the number of numbers less than and prime to N, 
is equal to a#—1L°-*(a—1)(6—1) ..; the question arises to find 
* Communicated by the Author. 
