18 G. F. BECKER FINITE STRAIN IN ROCKS. 



changes of length. Then, if x y are the original coordinates of any point 

 and / y its final coordinates these positions are connected by linear 

 relations, 



a! = (l + e)z+ by; y' = ax + (1 + /) y; / = (I + g) z; 



or, 



(1+/K-V . v _ (1 + e) y' - «*' . _^_ 



(1 + e) (1 +/) - ai " (1 + «) (1 +/) - «6 ' 1 + 1/ 



Here a, &, <?,/, <? are absolutely arbitrary and have the same value at 

 all points of the mass.* They are the coordinates after strain of par- 

 ticular points. Denoting x = 1, y == 1, z = 1, by (1,1, 1), points originally 

 at (1, 0. 0), (0, 1, 0), (0, 0, 1), are transposed to (1 + e, a, 0), (b, 1 +/, 0), 



(0,0,1+flr). 



When the strain is so small that the squares of the displacements are 

 negligible, a, b, e, /, g are to be treated mathematically as infinitesimal ; 

 consequently any formula in terms of this notation can lie converted 

 into the forms appropriate to small strain simply by neglecting powers 

 of a, b, e,f, g, higher than the first. 



Strain Ellipse. — The sphere x 2 + y z + z" = 1 is converted into an ellipsoid, 

 which is found by substituting for x, y and z their values in terms of the 

 accented variables. The section of this ellipsoid by the x y plane is an 

 ellipse with semi-axes A and B. Its equation is — 



j(l+/) 2 -f a , }a!"-2|&(H-/) + a(l + 0}^+{a + c) , + 6 s }^ f 



= {(l + e)(l+/)-a&}'. (i) 



When 6 (1 +/) 4- a(l + e) is a positive quantity the major axis of this 

 ellipse makes a positive acute angle with o x. Well-known properties 

 of the ellipse show that its area is the same as that of the circle — ■ 



*' 2 + V' 2 = (1 + c) (1 +/) — oft - A B, * (2) 



and that the axes may be found from the equation — 



(A ± By = I (1 -I- e) ± (1 + /) Y + (« + b)\ (3) 



*The letters e,/and g are used in the same sense as in Thomson and Taifc, Natural Philosophy, 

 but I have not found it convenient to use a and o as they are there employed. 



