288 
ME. GEOEGE H. DAE WIN ON THE INFLUENCE OE 
Since 
jj F(5, <p) sin QdQ d<p= 0, therefore a+b+c=0. 
If A be the moment of inertia of the body, after deformation, about an axis parallel 
to x, through x, y , z, 
A — A a — M.(y 2 + z 2 ). 
Now a varies as whilst M(y 2 -j- 2 2 ) varies as . But the greatest elevation or de- 
pression to be treated of is about two miles, whilst the mean radius c is about 
4000 miles ; hence ~ cannot exceed about 2 ifoo, and accordingly the term M(y 2 +5 2 ) is 
negligible compared to a. Whence A=A+a. 
In like manner, the terms introduced in the other moments and products of inertia 
by the shifting of the earth’s centre of inertia are negligible compared to the direct 
changes. Thus it may be supposed that the centre of inertia remains fixed at the origin, 
and that the moments and products of inertia of the earth after deformation are 
A-j-a, A+b, C-f-c, D, E, F. 
11. General Theorem with respect to principal Axes. 
A general theorem will now be required to determine the position of the principal 
axes after the deformation. 
Take as axes the principal axes of a body about which its moments of inertia are 
A, B, C. Let the body undergo a small deformation, which turns the principal axes 
through small angles a, (3 , y about the axes of reference, and makes the new principal 
moments A', B', C\ And let the moments and products about the axes of reference 
become in consequence A-}- a, B+b, C+c, D, E, F. Then it is required to find a, /3, y 
in terms of these last quantities. 
Let l, m , n be the direction cosines of any line through the origin, and let them 
remain unaltered by the deformation. Let I be the moment of inertia about this line 
after deformation. Let m-\-hm, n-\-hn be the direction cosines of the line with 
respect to the new principal axes. Then, by a well-known theorem, 
11= ym — j3n, hn = an — yl, hn=@l—ctm. 
Now 
I=(A-|-a)£ 2 -}-(B+b)m 2 + (C+c)w 2 — 2Dmn— e TEnl— 2F/m. 
But it is also equal to 
A! (I T hi) 2 -f- B '(m T &n) 2 - J- G(n-\~hny, 
and by substituting for hi, hm, hn, this is equal to 
A'l 2 A-B'm 2 +C'n 2 -2mn(a-lB , )a-2ln(A'-C , )[3-2lm(B' -A') y 
to the first order of small quantities. 
