ME. EOBEET MALLET ON VOLCANIC ENEEGY. 
I / O 
Lastly, we have to compare these results and. apply them to the actual pheno- 
mena of volcanic action on our globe. 
The first of these involves a mathematical investigation only, the last two rest upon 
two extensive series of experiments which have been made by the writer and are here to 
be detailed. 
82. First, then, the unsupported solid shell must crush by its own gravitation and that 
of the nucleus, if of any solid material known as part of our globe. 
Lagrange, in his ‘ Traite de Mecanique Analytique,’ cap. iii. sec. ii. (statique), “ Sur 
lequilibre d’une surface flexible,” &c. (Bertrand’s 4to edition, Paris, 1853), has given, 
though in an involved form and without proof, a theorem which is applicable to this 
question. This theorem has been reduced to simpler form by Professor S. Haughton, 
F.T.C.DA, who has applied it to a widely different subject from ours, and to whom the 
writer is indebted for having had his attention directed to its applicability to the 
present one. 
A proof of the theorem has been since produced by Professor II. S. Ball, of Dublin 
(Phil. Mag. vol. xxxix. pp. 107 & 108, Feb. 1870). The theorem may be thus stated : — 
If a curved surface (of the nature of a hollow shell or membrane) be in equilibrium 
when exposed to forces acting normally to the surface everywhere, then the normal, 
pressure at any point is equal to the force in the direction of the surface (or shell) 
at that point, multiplied into the sum of the reciprocals of the principal radii of 
curvature. 
The pressure may be internal (as in a blown bubble), producing tensions, or may be 
external (as in the case before us), producing pressures or thrusts in the direction of the 
surface or tangential to it; and the surface may be extensible or inextensible, but it 
is one into the consideration of which cross or shearing strains do not enter. 
83. Let P (fig. 7) be the normal pressure upon a 
unitof surface (square inch or mile) cut from a pair of 
intersecting ribbons of the curved surface, as a b and 
c d, at right angles to each other and of unit breadth, 
T the tangential thrust on any of the faces of the 
unit square respectively opposite (which, as being 
small in relation to the radii of curvature, may be 
considered as plane). 
Let the two radii of principal curvature (in ab and 
c d) be Oy and p 2 , then, as expressed in the theorem, 
P= 
I. 
T' having the same value. 
As regards the present application of the theorem, as the differences of ?1 and g 3 for 
our globe are very small (comparable with the difference between the polar and equatorial 
* And by Professor Miller in Ms ‘Hydrostatics ’ (Cambridge, 1831). 
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