162 Proceedings of Royal Society of Edinburgh. [sess. 
On the Directions which are most altered by a Homo- 
geneous Strain. By Prof. Tait. (With a Plate.) 
The cosine of the angle through which a unit vector p is turned 
by the homogeneous strain is 
Hence the required vector, its positions after the strain, and after 
a subsequent application of the conjugate strain, lie in one plane ; 
and the tangent of the angle between p and its first distorted 
position is half of the tangent of the angle between it and its 
doubly distorted position. 
When the strain is pure, the required values of p are easily 
found. Let the chief unit vectors of </> be a, /3, y, and its scalars 
g v 9y> 9s- Then the equation above gives at once three of the 
form 
(Read Dec. 7, 1897.) 
Sp<bp 
Tp.T(f>p ' 
This is to be a maximum, with the sole condition 
Tp = 1 • 
Differentiating, &c., as usual we have 
xp = — 2g>p Spc//</>p + <f> '4*9 . 
Operate by S .p and we have 
— x = — Spcj>p S prf>'<f>p ) so that 
4>p <f>'c ftp 
P= - 2 rrr- ; • 
iSpcfip i6pcf>'(f)p * 
There are two kinds of solutions of these equations. 
First. Let the first factor vanish in two of them, e.g., 
S /3p — 0 , Syp = 0 , or p = a. 
