of Edinburgh, Session 1879-80. 403 
solution. Let <j£> be divided into parts, one self-conjugate, the other 
not, then 
(ft = + Y.e , 
and the given equation may be written 
ttp + Yep = Xp . 
Hence S .p j (w - x)a + Yae | = 0 
whatever be a. Let a, /3, y be the principal unit-vectors of the 
pure strain &f, and a , b , c (in descending order of magnitude) the 
associated scalars. Then the equation for x is, at once, 
S.^(« - x)a + Y ae^(b — x)j8 + Y/3e ^ (c - x)y + Yye ^ = 0. 
This may be written as 
(x - ci)(x - b)(x - c) - efx + S.Ue ^Ue) = 0. 
Thus the problem is reduced to finding the limiting value of Te, for 
any given value of Ue, so that the above equation may have all its 
roots real. This leads by the ordinary methods to a cubic in Te 2 , 
but the expression is rather complicated. 
For variety let us adopt a graphic method. It is obvious that 
the extreme values of - S.Ue roLJe are a and c. 
Let the curve represent the equation 
y = (x - a){x - b){x - c), 
and let OD represent any assumed value of - S.Ue SfUe. D must 
lie on the finite line AC. From D draw, as in the figure, a tangent 
3 c 
VOL. X. 
