ON SLATY CLEAYAGE AND ALLIED ROCK-STRUCTURES. 827 
By observing the distortions 2 and as for two places in the same locality 
Pp 
where the bedding makes angles ¢ and ¢’, respectively, with the cleavage, 
we can calculate the ratios of the axes of the strain ellipsoid. For it is 
easy to show that, from equation (ii.) and the corresponding one, we 
obtain 
\'=(58 P sing!) (5 si sin» ) iad 
—) =/{—sm¢-+- sin ¢ — smn @¢@— — sim Ai ll. 
(<) di order uinans Salles aoc ee 
sin (> +9") sin (p— 9") 
2 
(:) =(° cos ¢’+ ae cos ) (2 cos g! — Ay cos ) Od Mee) 
c p p \e p 
sin (p +9’) sin (p—¢’). 
The practical question is, then, how to determine the distortion i for 
any position of the bedding-plane by observations of deformed fossils in 
that plane. Let us call the length of a fossil, such for instance as a 
trilobite, that line about which, in an undistorted specimen, there is 
bilateral symmetry, and let the breadth be measured along a line which 
before distortion is at right angles to the length. The length and breadth 
of a distorted fossil will not in general be at right angles to each other: 
they will be so only when one of them lies in the direction of dip, and 
consequently the other in the direction of strike. In this special case we 
may profitably adopt Professor Haughton’s method, viz., comparing the 
relative dimensions of the distorted fossil with the known relative 
dimensions of the species when undistorted. If the length of the fossil 
lie in the direction of dip, the ratio of its length to its breadth, divided 
by the ratio of length to breadth in an undistorted specimen, will give 
; ; if the breadth of the fossil lie in the direction of dip, the same calcu- 
lation will give © {fon the same slab of rock occur specimens in the 
two positions, we can estimate the distortion without knowing the 
undistorted form of the species; for if we divide the ratio of length to 
breadth of a specimen in the former position by the ratio of length to 
2 
breadth of a specimen in the latter, we get at once (; 
A single specimen of a fossil whose length and breadth lie oblique to 
the dip and strike, is sufficient to determine the distortion without pre- 
vious knowledge of the form of the species when undistorted. For if a 
and (3 be the angles which the length and breadth make with the direction 
of dip, it is readily proved that 
(5) = tana tan Wi Caen nd eee he SM 
This method is readily applied, and if the length and breadth make 
considerable angles with the dip, it is very accurate. It is tantamount 
to that employed by M. H. Dufet,! who, however, makes use of geome- 
trical constructions. 
( 1 « Déformations des Fossiles, &c.,’ Ann. de ? Ecole Norm. Sup., sér. 2, t. iv. p. 183 
1875). 
