THE MECHANICAL INTERPRETATION OF JOINTS 17 
acute angle of shearing planes faces the direction of the com- 
pressive stress. We may now extend the law by adding, that in 
ductile substances it is the obtuse angle that faces the direction of 
the compressive stress. 
Before attempting to apply the law to any specific case, there- 
fore, we must decide whether the material under the given con- 
ditions had the properties of a brittle or those of a ductile substance. 
On the other hand, when the direction of the greatest principal 
stress is known, the position of the joint planes produced by it may 
be used to determine the degree of ductility which the material 
possessed at the time of shearing. 
All the cases so far discussed involve irrotational strains only. 
The arrangement of shearing planes due to rotational strains, as 
illustrated by Leith’s wire-netting model and discussed in his book 
on Structural Geology, is, of course, only possible in ductile sub- 
stances, as a glance at the angle of the shearing planes will show. 
This model has, however, been applied successfully to some 
striking cases of jointing in quartzites. 
We may approach the problem involved in these interesting 
cases by turning to ‘an illustration in Van Hise’s “Principles of 
North American Pre-Cambrian Geology,” page 652.1 Figure 131 
shows layers of quartzite alternating with thin beds of more slaty 
character. The harder beds are traversed by two systems of 
intersecting joints, both forming angles of 50°—70° with the bedding 
planes, that is, forming acute angles of approximately 60° facing 
the bedding planes. 
In the intercalated slaty beds, however, only one of these two 
joint systems is developed. It consists of more numerous joints 
inclined but 20° or less to the bedding planes. If the comple- 
mentary symmetrical set were developed, the angle formed by the 
two systems facing the bedding planes would be 130° in these less 
brittle slaty beds, instead of 60° as in the brittle purer beds. 
From this relation of the shearing angles in the two types of 
rock it is evident that the joints in the more brittle beds are due to 
the normal component of the stress acting on the beds. They 
t Sixteenth Ann. Rept. U.S. Geol. Survey, Part I (1896), p. 652. See also 
C. K. Leith, “Rock Cleavage,” U.S. Geol. Survey, Bull. 239 (1905), Pp. 123, Fig. 37. 
