Mr. A. Cayley ona Surface of the Fourth Order. 491 
principal angle of incidence, and the index of refraction, to de- 
cae the value of A, and consequently by (18) the value of 
e dv. If now it be supposed that p can be approximately de- 
I — 
termined by a , the thickness of the intermediate layers 
= 
can be deduced from the experiments. 
In this way we have a new means of testing the theory, since 
this requires that the thickness in question should be small and 
positive. The calculation from Jamin’s experiments shows that 
this is actually the case, though, as might be expected, no great 
degree of exactness can be thus attained in the determination 
of these quantities. Ihave found that the thickness of the layers 
a the case of the bodies experimented on lies between 55 and . 
rho of the length of a wave. 
It appears, therefore, that the result of Jamin’s experiments. 
can be completely explained on the simple supposition of an ex-. 
ceedingly thin stratum of intermediate layers in which there is 
a gradual change of refractive power, a supposition which we. 
have obyiously- more right to make than to omit. 
- Copenhagen, June 28, 1860. 
LXXII. On a Surface of the Fourth Order. 
By A. Cavity, Esq.* 
iP! A, B, C be fixed points; it is required to investigate 
the nature of the surface, the locus of a point P such that 
AAP + ~BP+rvCP=0, 
where A, p, v are given coefficients; the equation depends, it is 
clear, on the ratios only of these quantities. 
_ The surface is easily seen to be of the fourth order; it is ob- 
viously symmetrical in regard to the plane ABC; and the section 
by this plane, or say the principal section, is a curve of the fourth 
2 hee the locus of a point M such that 
AM + pBM +7CM=0. 
The curve is considered incidentally by Mr. Salmon, p. 125 
of his ‘ Higher Plane Curves ;’ and he has remarked that the two 
circular points at infinity are double points on the curve, which 
is therefore of the eighth class. Moreover, that there are two 
double foci, since at. each of these circular points there are two 
tangents, each tangent of the one pair intersecting a tangent of 
the other pair in a double focus; hence, further, that there are 
* Communicated by the Author. 
