550 Royal Society :— 
tion, I have supposed the ice under consideration to be porous, and 
to contain small quantities of liquid water diffused through its sub- 
stance. Porosity and permeation by liquid water are generally 
understood, from the results of observations, and from numerous 
other reasons, to be normal conditions of glacier ice. It is not, 
however, necessary for the purposes. of my explanation of the plas- 
ticity of ice at the freezing-point, that the ice should be at the outset 
in this condition; for, even if we commence with the consideration 
of a mass of ice perfectly free from porosity, and free from particles 
of liquid water diffused through its substance, and if we suppose it to 
be kept in an atmosphere at or above 0° Centigrade, then, as soon as 
pressure is applied to it, pores occupied by liquid water must in- 
stantly be formed in the compressed parts in accordance with the 
fundamental principle of the explanation which I have propounded 
—the lowering, namely, of the freezing- or melting-point by pressure, 
and the fact that ice cannot exist at 0° Cent. under a pressure exceed- 
ing that of the atmosphere. I would also wish to make it distinctly 
understood that no part of the ice, even if supposed at the outset to 
be solid or free from porosity, can resist being permeated by the 
water squeezed against it from such parts as may be directly sub- 
jected to the pressure, because the very fact of that water being 
forced against any portions of the ice supposed to be solid will 
instantly subject them to pressure, and so will cause melting to set 
in throughout their substance, thereby reducing them immediately 
to the porous condition. 
Thus it is a matter of indifference as to whether we commence 
with the supposition of a mass of porous or of solid ice. 
“On the Comparison of Transcendents, with certain applications 
to the Theory of Definite Integrals.’’ By George Boole, Esq., Pro- 
fessor of Mathematics in Queen’s College, Cork. 
The following objects are contemplated in this paper :— 
Ist. The demonstration of a fundamental theorem for the sum- 
mation of integrals whose limits are determined by the roots of an 
algebraic equation. 
2ndly. The application of that theorem to the comparison of 
algebraical transcendents. 
3rdly. Its application to the comparison of functional transcendents, 
i. e. of transcendents in the differential expression of which an arbi- 
trary functional sign is involved. 
4thly. Certain extensions of the theory of definite integrals both 
single and multiple, founded upon the results of the application last 
mentioned. 
In the expression of the fundamental theorem for the summation 
of integrals, the author introduces a symbol, ©, similar in its defini- 
tion to the symbol employed by Cauchy in the Calculus of Residues, 
but involving an additional element. The interpretation of this 
symbol is not arbitrary, but is suggested by the results of the inves- 
tigation by which the theorem of summation is obtained. All the 
general theorems demonstrated in the memoir either involve this 
symbol in their expression, or are immediate consequences of theorems 
into the expression of which it enters. 
