and the Distortion of Fossils, 411 



examine what effect the angle between the planes of cleavage and 

 bedding produced on the amount of distortion. The following 

 law expresses the result of my measurements : — 



II. The distortion of fossils produced by cleavage — estimated in 

 a given direction^ such as parallel to the intersection of the planes 

 of cleavage and bedding — varies with the angle between these planes, 

 being greatest when the angle is greatest, and least when the angle 

 is least. 



The distortion depending upon the angle between the planes 

 of cleavage and bedding may be calculated numerically from the 

 following expression : — 



A= VPsin2</, + Qcos^^, (2) 



in which A denotes the distortion parallel to the intersection of 

 cleavage and bedding, </> the angle between these planes, and 

 P, Q numbers ; P being always greater than Q. The mode of 

 determining the numbers P and Q will be presently explained. 

 In addition to the two laws of distortion already mentioned, 

 there is a third law, which is so easily observed that it has forced 

 itself on the notice of every person who has paid even a slight 

 attention to this interesting subject. It maybe stated as follows : — 



III. The compression in a cleaved rock is greatest in a direction 

 perpendicular to the planes of cleavage. 



Having ascertained satisfactorily the truth of the laws of dis- 

 tortion just given, I proceeded to form a general theory by the 

 aid of the following considerations. 



It has been beautifully proved by Cauchy, that if a perfectly 

 homogeneous, uncrystalline mass be subjected to pressures, twists, 

 strains, &c. of any kind, it will settle into a condition of equili- 

 brium, subject to the following law : — Imagine a small sphere in 

 the mass before distortion ; this sphere will become an ellipsoid 

 after distortion ; and if the distortions in the principal planes be 

 known, we can calculate the forces which produced the distortion 

 in the mass. 



This hypothetical problem in mechanics, solved by Cauchy, 

 is illustrated by the phsenomena of cleaved rocks ; and since, by 

 the measurement of the distortion of the fossils, we can ascertain 

 the amount of distortion in each plane of bedding, we can readily 

 obtain data for calculating the axes of the ellipsoid of distortion, 

 and so obtain an idea of the direction and magnitude of the force 

 which has cleaved the rocks and distorted the fossils. The mode 

 of determining the axes of the ellipsoid is as follows : — Since, by 

 the first law, the intersection of the planes of cleavage and bed- 

 ding is an axis of the ellipse of distortion in all planes of bedding, 

 it must be an axis of the ellipsoid. Also it is certain, from the 

 third law, that the compression is a maximum perpendicular to 



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