832 REPORT—1885. 
course, less than unity). This is in accordance with the rules stated 
above, as deduced from observation. To obtain a numerical test, suppose 
that, during the process which results in the cleavage structure, the slate 
is reduced to half its former volume, while the grit remains unchanged in 
bulk, so that m=4. Suppose, too, that the distortion of the slate is such 
that a= 6c; then equation (vili.) becomes 
8 143 
2q( se a wot” cosine » old Spe 
cot 29 gn Coto {ao 27 93 (ix.) 
from which we can find 9g’ when ¢ is known. Thus, if ¢=45°, we get 
¢’ = 64° 12’, and therefore the deflection of the cleavage planes would be 
19° 12’; or if 6=30°, we get ¢’ = 46° 45’, and so a deflection of 16° 45’. 
These figures agree very well with observation. 
As an approximate rule, when the slate is strongly cleaved, we may 
use the equation 
ARP hehe ieee Oe Jee 11. 2.) acum elon hiepobeeeel 
22 
which results from neglecting the quantity =f in equation (viii.), 
a 
Another peculiarity of contorted gritty bands intercalated among 
slate rocks may be investigated by means of the above considerations— 
viz. the fact that a thin band of this kind is found to be thickened at the 
crests and troughs of the undulations (where the band is cut perpen- 
dicularly by the cleavage), and more attenuated at the intermediate 
places. In order to find a’ and c’ from equation (vii.), we may make use 
of the relation 
1, 1 _ cos’ + m?sin®@ , sin? + m?cos’ 
pie lactaalaon aged } bck & dol Gas Ae 
a? ¢ a c 
which reduces to 
a/? + ¢/? = { (1+m?) (a?+c?) —(1—m?) (a? —c?) cos 2 o} p(x) 
This equation, together with the one 
. (xii) 
suffices to determine a/ and c’ ; and since the values thus obtained depend 
upon ¢, the distortion of the gritty band will vary in different parts. It 
is greatest, as appears from equations (xi.) and (xii.), where cos 2¢ is 
least—z.e. where ¢ = 90°, or at the crests and troughs of the folds. At 
these places the thickness of the band is increased in the ratio ws (=) ; 
which, with the numerical data assumed above, is 3°46. These numerical 
results accord very closely with what I have noted in the slate districts of 
North Wales. 
Other peculiarities at the junction of slate-rock with a more gritty 
bed may be explained in a similar manner. One of these characters was 
mentioned and figured by Dr. Sorby in the Ilfracombe section already 
mentioned. On approaching the gritty band, which shows a series of 
contortions, the cleavage-planes of the slate are seen to curve slightly 
away from the crests of the folds, in a manner suggestive of ‘stream- 
