272 Mr. G. B. Jerrard’s Remarks on 
the dispersive and extrusive powers of the medium are large 
that its inapplicability becomes manifest. 
The exponential law, on the other hand, entirely overcomes 
the difficulty arising out of the irrationality; because it shows 
that in each medium there is, dependent on the proportion which 
the irrationality bears to the dispersive power, a certain exponent 
for the normals at which the extrusions attending the irrationality 
are reduced to a minimum, and that with this exponent the in- 
dices may always be obtained from two constants,—each index 
being then reduced to two terms, one of which, ¢,, is constant 
for the medium temperature and exponent ; while the other (a,) 
corresponds to a further shortening of the wave-length within 
the medium, which is constant for each wave, and so inversely 
proportional to the primary wave-lengths of the normals, with 
this particular exponent applied to them, the formula for each 
n 
a8 
index being w= 
ad an 
En 
[To be continued. } 
XXXV. Remarks on Mr. Harley’s paper on Quintics. 
By G. B. Jerrarpt. 
N the ‘ Quarterly Journal of Pure and Applied Mathematics ’ 
for last January, there is a paper by Mr. Harley “On the 
Theory of Quintics,” respecting which | am induced to offer a 
few remarks. 
1. On comparing the results at which he has arrived, 
2° 4 5QE? + / {E(E8—108Q°*)}.¢—5Q*=0, . (a) 
bata tote + typ 2 (Oo ee ye he a eed 
in his explanation of Mr. Cockle’s ‘Method of Symmetric Pro- 
ducts,’ we may easily perceive (for Q, E are the coefficients of 
the trinomial equation in w with which he sets out) that the 
method in question is in general not applicable to equations of 
the fifth degree. 
For as the equation (w,) belongs, according to art. 8 of Mr. 
Harley’s paper, to a class of equations of the sixth degree, solved 
by Abel, the roots of which, as is well known, do not involve 
any radical higher than a cubic, it is manifest, from (4), that 
* The refractive index of any medium at a given temperature, for white 
light, may be found very accurately from the above formula by making 
A=0-933494, the length of the mean wave, in relation to that correspond- 
ing to the fixed line B as unity. 
+ Communicated by the Author. 
