FORMULAS FOR STRAIN ELLIPSE. 19 



The third axis of the ellipsoid is C= 1 + g> Iff is the length of any- 

 one of the axes, A, B and G are the three roots of the cubic — 



( n -AUv-B)(ji-0) = 



1 f.» 



| 9 — (1 + flr) | [f 2 --f !/ (1 + « + 1 + /) 2 + (a -" ^) 2 + 



(l + «Xl+/)-«*} = 0. .(4) 



The volume assumed after distortion by the unit cube may be called 

 /,3 = A BC=(1 + g) •[ (1 + «)(!+/)- a& } . (5) 



A 3 , and — 



Rotation. — The limitations of this discussion imply that the plane of 

 A G can only revolve about G, so that the position of this plane is de- 

 termined when the position of A is known. The angle which A makes 

 with o x is, say, v, and this angle can immediately be inferred from (1) 

 by a well-known formula which gives — 



a 2 — &' + (l +/)'--( l+«)' 



Since the plane i? is at right angles to that of A G, its position follows. 

 To find the position which the same material lines A and B occupied in 

 the unstrained mass, it is convenient to remember that they must have 

 been at right angles to one another before strain as well as after it ; for 

 mere rotation changes no angles, and irrotational strain is by definition 

 a deformation in which the ellipsoidal axes maintain their direction. 

 Hence, if// was the angle which the fiber A made with o x before distor- 

 tion, its equation was y\x == tan //, and by the displacement formulas 



y' a + (1 + /) tan ft 



tan v = — = -rz — : — s — ,—j—, 



x (1 + e) -f- b tan ft 



The angle which the other axis made before strain was ,»■ + 90°, so that 

 tan (fi + 90°) = — cot ft, while after strain it becomes v -f- 90°. Hence — 



tan + 90°) = ?~ ( l +f lZ* = ~ «* '■ 



v J ( 1 -\- e) — b cot fl 



