MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
537 
the rupturing forces will be perpendicular to the neutral surface, and therefore, from 
a well-known property of that surface, and 4 will be the perpendicular distances of 
the centre of gravity of the section of rupture, from the outer and inner surfaces of 
the shell. 
The force developed upon the element c?L, opposed to the compression or extension 
si 
of the material of the band at the distance 4, will be and its moment about the 
line formed by the intersection of the neutral surface with the section of rupture will 
be 7 
h 
Let Ta represent the radius drawn from the centre of the spheroid to the intersec- 
tion of the neutral surface with the section of rupture ; then the moment of the pres- 
sure Pi between the section of rupture and the parallel of & will be sin (^— ^ 2 )^L', 
where 6 stands for the latitude at any point of the band subjected to tension, and 
dU an element of the area of the shell’s inner surface. If we represent by the 
equatorial axis of the shell’s inner surface, co^ the arc intercepted between the planes 
bounding the band, the equatorial axis of the neutral surface, and make 
we shall have 
dl.= 
Wi^i cos fl 
1 — sin^fl’ 
r=h .^\/ 1 — sin^^. 
As the sum of the moments of the pressures exerted on the band at either side of 
the section of rupture, when the shell is in the state bordering on rupture, must be 
equal to the sum of the moments of the cohesive forces at the same section, we shall 
have 
sin (9 — cosfl<^9 
(1 — sin^fl)^ 
But from article 3, Pi=/i(/(i-j- cos^^), making for brevity 
^ = (/-/i ) J h,= 
TT 
and remembering that here of that article is represented by 6. Hence 
fsPdLi + cos^ S) sin(fl — flg) cos 
J e, (l-s"sin2fl) i/l_e2sin2 9 
I 
=Ai COS cos ^ sin ^ (1 — sin^^)“^fl?^— ^i sin ^a/'cos^ ^(1 — £^ sin^ d)~^dS 
-f cos cos® ^ sin ^ ( 1 — £® sin® &)~^dd — sin cos'* ^ ( 1 — £® sin® dy^dd. 
The integrals multiplied by cos 4 can be easily-found by ordinary methods, but 
the two multiplied by sin 4 cannot be given in a finite form. But from the nature 
of the problem one of the limits at least of the integrals must be an independent 
variable ; hence the integrals multiplied by sin can be determined only by deve- 
