ON SLATY CLEAVAGE AND ALLIED ROCK-STRUCTURES. 831 
Dr. Sorby ! long ago noted similar laws, and gave his explanation of 
the phenomenon, which is briefly this. Since the grit yields less than the 
slate to the compressing force, the total voluminal compression is greater 
for the slate than for the grit. But near the junction of the two rocks the 
change of dimensions in the direction parallel to the bedding must be 
the same for both. Consequently, in the direction perpendicular to the 
bedding, the slate undergoes a less expansion (or greater compression) 
than the grit; and the cleavage planes, which are in each rock perpen- 
dicular to the direction of greatest compression, will therefore be less 
inclined to the bedding in the slate than they are in the grit. 
Professor Tyndall ? draws quite different conclusions from the pheno- 
mena in question, connecting the deflection of the cleavage planes with 
the progressive development of the structure concurrently with the con- 
tortion of the beds. Referring to a contorted gritty bed intercalated in 
slate-rock, he says: ‘When the forces commenced to act, this inter- 
mediate bed, which, though comparatively unyielding, is not entirely so, 
suffered longitudinal pressure ; as it bent, the pressure became gradually 
more lateral, and the direction of its cleavage is exactly such as you 
would infer from a force of this kind; it is neither quite across the bed, 
nor yet in the same direction as the cleavage of the slate above and below 
it, but intermediate between both.’ 
The latter statement seems rather obscure, and raises the question 
how far the production of the cleavage structure and the concomitant 
contortion of the strata can be regarded as contemporaneous. Dr. Sorby’s 
exposition seems sufficient, and is a necessary consequence of the me- 
chanical theory. To compare the theory with the phenomena we may 
proceed as follows. 
Let a, b, c be the semiaxes of the strain ellipsoid for the slate, and 
a’, 0’, c’ for the grit, and let ¢ and 9’ be the angles between the bedding 
and cleavage for the two rocks respectively. Referred to axes re- 
spectively parallel and perpendicular to the bedding, the Cartesian 
equation to the ellipse in fig. 6 is, for the slate, 
(cos ty sing)” , (ycos¢ SEU ER OREIIS Ihe” Bay 
a c 
UP 
The change of volume is in the ratio a for the slate, and ra for the grit, 
and, consequently, if we call the ratio ac: a/c’, m, the equation to the 
ellipse for the grit is 
a aiht ahs eee ge 
ee) mes xsin g)?_ Ee 
a c 
By the usual rule for finding the axes of an ellipse, we may deduce that 
cot @ (ma? —c?) — tan ¢ (a?— mc?) 
a 
cot 2 9’ = Im (@—2) 
From this last equation it may be seen that if ¢ = 0, 9! =0;"land if 
$ = 90°, ’ = 90°; but in any other case ¢/ is less than @ (for mjis, of 
1 Edinb. New Phil. Journ., cit. (1853). Cf. Professor H. D. Rogers, Trans. Roy. 
Soc. Edinb., vol, xxi. p. 449 (1856). 
2 Phil. Mag., 4th ser., vol. xii. p. 43 (1856). 
