MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
529 
Hence, making a'=l, 
where represents a number which always differs but little from 1. The coefficient 
of a\ being small, it is evident that the numerator of this expression cannot vary 
rapidly with a,. 
From article 12, it is evident that must always either increase or decrease as 
increases ; in order to know how it varies, it will therefore be only necessary to ex- 
amine its value for any two values of a,. 
Let a,= 1, then 
(a) being determined by observation, and K having the value assigned to it in art. 8, 
Part I. It will appear that the greatest value of K which can be assigned will make 
e a little less than 
When tfi=0. 
ff'i and Vi being nearly equal to unity ; and and a not much differing, e' will not much 
differ from m, and consequently it will be greater than e. This remark can be veri- 
fied by actual calculation, when we shall have obtained the numerical values of the 
quantities contained in the preceding expression. 
V. THE PRINCIPAL MOMENTS OF INERTIA OF THE EARTH. 
14. The dependence of all the phenomena of the earth’s rotation upon its principal 
moments of inertia, and the resulting connexion which thus subsists between these 
phenomena and the internal constitution of the earth, renders a complete investiga- 
tion of their values of some importance for the objects to be fulfilled by the present 
memoir. 
Let the moment of inertia with respect to the axis of rotation be C, and A and 
B those for the other two axes. Let C,, &c. represent the moments of inertia of the 
solidified spheroid included within the shell’s outer surface, Cg, &c. the moments of 
inertia of the spheroid included within the shell’s inner surface, and Cg, &c. the mo- 
ments of inertia of the nucleus, then 
C=Ci — C2+C3, B=Bi — Bg-j-Bg, A^A, — Aj+Ag, 
and representing an element of the earth’s mass by dm, 
C=f{3^ -\-y^')dm, ^~f{x^-\-z^)dm, A=f{y^-\-z^)dm, 
X, y, z being the rectangular coordinates of the element dm. In finding C,, &c., Cg, &c., 
Cg, &c., different values for dm must in general be introduced in the foregoing expres- 
