MAGNETISM AND TWIST IN IKON AND NICKEL. 387 



and hence the equation to the strain ellipsoid is 



(1 - 2o-)X 2 + (1 - 2a) Y 2 + (1 - 2?«r)Z 2 - 26xYZ + 2ByXZ = 1 . 



By choosing the y or Y axis along the line from the centre of the section to the point 

 x y, we get this equation in the somewhat simpler but no less general form — 



(l-2o-)X 2 +(l-2o-)Y 2 +(l-2sr)Z 2 +2%XZ = l .... (2) 



It is evident the Y axis is a principal axis of the strain ellipsoid ; and in finding the 

 others we may confine our attention to the plane XZ. 



Let X /a be the direction cosines of a principal axis in this plane, and r the correspond- 

 ing radius, then 



- 2<7\ 2 - 2sr M 2 + 2dy\ft = \-l 



\ 2 + M 2 = l. 



Differentiating and remembering that 1/r 2 is either a maximum or minimum, we get, on 



reduction, 



X_ By _ 2n-p ,„ 



fx 2<r-p~ By ' ' w ' 



where 



, 1 



and satisfies the equation 



p 2 -2p(T*+a) + 4>zro— (By) 2 = 0, (4). 



Now let r x r 2 be the maximum and minimum values of r, then we may write 



r 1 2 = l + 2ar 1 r 2 2 = l + 2 ( r 1 



1-^ = 2^ 1-^ = 2^, 



where m^ and <r l are the principal elongations, and 2^ 2^ are the roots of equation (4). 

 Hence 



4z<r 1 a- 1 = 4sr<r — (By) 2 

 from which we find easily, 



2ar = w x + o- 1= b sJi^-a-.Y-iBy) 2 



2a =^+0-^ J(*-<r x T-(e y y 



The equation of the strain ellipsoid, referred to its own principal axes, may then be put 

 in the form 



(l-2w 1 )X 2 +{l-(7«r 1 + o- 1 )± V(*i-<Ji) 8 -(fy){ 2Y +(l-2o- 1 )Z 2 = l . 

 To find the angle which the major axis of this ellipsoid makes with the axis of z, we 



