AND COEWALL, AND THEIE MECHANICAL OEIGIN. 
411 
I shall therefore direct my attention at present to the phenomena of Secondary Joints, 
which have been shown to take place along directions inclined at about 30°, on each 
side of the primary direction of the original joints. Let ^ 
xy, x'y' be the directions of two primary parallel joints, 5 
and let a force P, acting horizontally, tend to produce r . 
a fracture in some unknown direction, ab, making an x > y' 
angle 0 with the directions xy. x'y 1 ; if the band of rock 
were capable of bending, this direction would be at right angles to the direction xy ; 
but if, from the pressure of superincumbent rock masses, or other cause, it be prevented 
from bending, it will fracture along the line ab, which is the plane of easiest fracture, 
and whose direction 0 is thus found. 
Let K denote the coefficient of cohesion of the rock, l the length of ab, and h the 
perpendicular distance between xy and ody', and let p = tan X denote the coefficient of 
friction of the rock. 
The mechanical condition of equilibrium is evidently as follows : — 
P cos 0=z'Kl-\-y J ¥ sin 0, 
or K i 
?cos0=£ Q+[ ,T>sm0 ( 1 ) 
Differentiating this equation, regarding P as a minimum and therefore (ZP=0, we find 
— Psm0= — ^ 5 j-+/aPcosA (2) 
Equating the values of P deduced from (1) and (2), we find 
sin 0 (sin 0-{-yj cos 0)=cos 0 (cos 0 — / sin 0) ; 
from which follows 
cos 20=^ sin 20 (3) 
or 
cot 20=tanX ; 
and, finally, 
20+X=9O° (4) 
The values of X, determined for various kinds of rock, limestone, sandstone, and slate, 
vary from 30° to 38°, thus giving for 0 corresponding values from 30° to 26°. These 
values agree very well with the angles between the systems of Primary and Secondary 
joints determined by the preceding observations in different localities ; and I believe that 
the mechanical cause I have assigned is the true cause of these joints. 
From this it follows that a single hypothesis as to the direction of a system of forces 
is sufficient to account for the existence of three conjugate systems of joints, involving 
six directions ; and it also explains the rudely hexagonal jointing of many rocks, if the 
coefficient of friction be such as to render the angle 0 nearly 30°. 
Many other consequences flow from the preceding investigations, into which I have 
not time to enter, but which may be readily found and turned to practical use by the 
field geologist who prefers one hypothesis to many. 
