. 
\ 
eS 
ON SLATY CLEAVAGE AND ALLIED ROCK-STRUCTURES. 823 
here developed, of all planes passing through b (fig. 5), the facility of 
splitting is a maximum for that plane which also passes through a, and a 
minimum for that which passes through c. It is easy to see, from con- 
siderations precisely similar, that, of all planes passing through c, the 
facility of splitting is a maximum for that passing through a and a mini- 
mum for that passing through 6: and, of all planes passing through a, 
the facility of splitting is a maximum for that passing through b, and a 
minimum for that passimg through c. Summarising these conclusions, 
we may say that the plane through a and bd gives a maximum, and that 
through } and ca minimum among all other planes in the rock, while 
the plane through a and c gives a maximum among all planes through ¢, 
and a minimum among all those through a. With respect to these 
results, two remarks may be made. 
In the first place, the principal diametral plane of the strain ellipsoid 
which passes through a and } is the cleavage-plane par excellence; but 
the facility of cleavage along other planes nearly parallel to it is nearly 
as great, and it is evident that under suit- 
able stresses the rock will split along any Fie. 5. 
plane making a small angle with the true 
cleavage direction. This is not a truism, 
but a consequence of the fact that the fa- 
cility of splitting along the true cleavage 
is a maximum in the proper mathematical 
sense, and varies continuously from that 
plane ; the same would not be true of the 
crystalline cleavage of minerals. 
The other point to be noticed is that, 
on this theory, there can be but one true 
cleavage direction: a plane parallel to the 
‘side,’ z.e., the plane passing through a and 
¢, is not a cleavage-plane in the sense just 
indicated. For although the facility of 
cleavage along such a plane is a maximum 
for all planes passing through c¢, it is a 
minimum for all planes passing through a. A similar argument holds 
good on Mr. Sharpe’s theory of the constitution of slate-rock, and his 
‘secondary cleavage’ seems to be no more than the properties of ‘side’ 
referred to below, in Section VI. It has been suggested ' that a second 
cleavage might arise from the action of a second and subsequent lateral 
pressure, operating in a different direction from the first one. But this 
is not the case, for after any combination of uniform compressions and 
expansions of the rock, the strain surface will still be an ellipsoid, and 
the line of argument indicated above will hold for this final or resultant 
strain ellipsoid. 
In the case of the fibrous or ‘linear’ cleavage mentioned above, a pheno- 
menon of only local occurrence, if we suppose b = ¢, the facility of cleavage 
along all planes passing through a will be the same, and greater than 
that along any other plane. In the case of Professor Haughton’s strain 
ellipsoid, in which a = 8, it will be seen that the facility of cleavage will 
be the same for all planes passing through c, which is incompatible with 
the distinctive properties of the ‘side’ and ‘end,’ recognised by the 
. Quarrymen in all the best roofing-slates. 
1 Quart. Journ. Geol. Soc., vol. v. p. 116 (1849), &c. 
