540 
MR. HENNESSY’S researches IN TERRESTRIAL PHYSICS. 
fraction, j will be small compared with or 4 ; hence in general, when the shell is 
thin, 
21. The function in which 6^ is to be made a minimum is by art. 19, 
(A)- [(fli) sin (93 + 02) sin (03-^2) + («2) sin 2(93 + 62) sin 2(63-62) + (flg) sin 8(63 + 62) sin 8(63-62)] 
+ [(Jj) cos (63 + 62) sin (63—62) + (62) cos 2(63 + 62) sin 2(63— 62) + (^3) cos 8(63 + 62) sin 8(63—62)] tan 63 
+ (ci)( 63— 62) tan 63 = 0 [(A) being a function of «j]. 
or 
(A) — [(flj) sin (63 + 62) — (6 1) cos (63 + 63) tan 63] sin (63- 63) — [(flg) sin 2(63+62) — (^3) cos 2(63 + 63) tan 63] sin 
— («3) [sin 8 (63 + 63) — cos 8 (63 +63) tan 63] + (ci)(63— 63) tan 63=0 42.) 
From this expression we can deduce the following conclusions : — 
TT 
1st. If ^2 be less than 63 cannot be zero unless (A)=0. 
2nd. If ^2=^= ^3, cos (^3 -1-^2) = — Ij cos2(^>3+^2)=0, cos 3(^3+^2) = — 
and consequently 
(A) = (&i) + (&3)- (c,) = g[^l— + 
A,=|(l-8(A))-i. 
3rd. In order that in this case 11= A, we must have (A)=gf 1— 
From article 20, 
I / §2(1 + 
0 
4s 
(A): 
(1 — 2e,)(l +ei)^(l— «i)«i^ 
The value obtained for the numerator of this expression, shows that for no con- 
ceivable value of (A) can vanish ; and therefore from the first of the preceding con- 
clusions, it may be deduced that if the shell be fixed under any parallel, it must be 
also fixed under some other parallel at the same side of the equator and at a distance 
from it, depending on the cohesive strength of the material of the shell. Conse- 
quently in the case of ^2<C 2 ^ zone of least disturbance should exist. In general it is 
evident that the value of (A) ^supposing which would make ^2 a maximum, 
must be extremely small ; consequently, from the second and third conclusions, if 
^2=-, or if no zone of least disturbance exist, IT must be considerably greater than h. 
