1897-98.] 
Prof. Tait on Homogenous Strain. 
163 
Then the remaining equation is satisfied identically, because its 
second factor becomes 
1 - Sh H • whence it 1 — 1 . 
Ui 9i\ 
Thus, as we might have seen at once, the lines of zero alteration 
(minima) are the axes of the strain. 
Second. Let the second factor vanish in two of the equations, 
i + 1 + ^_^L =0 . 
IS pcf>p Sp</> 2 p 5 ISp</>p Sp</> 2 p 
These give at once 
Sp<f>p = 
so that 
W 3 
92 + 3-3 
u z = 
, Sp<f> 2 p = - fjjj, ; 
4 Ms 
Oa + fl's ) 2 
In this case it is evident that we have also 
Sap = 0 « 
[In fact, neither the first factors, nor the second factors, in the 
three equations, can simultaneously vanish : — except in the special 
case when two of g v g 2 , g 3 are equal.] 
Of the three values of u 2 just found, the least, which depends 
upon the greatest and least of the three values of g, gives the single 
vector of maximum displacement: — the other two are minimaxes, 
corresponding to cols where a contour line intersects itself. 
(Read February 21, 1898.) 
The self-intersecting contour-lines, corresponding to 3, 2, 1 as 
the values of the ps, were exhibited on a globe ; whose surface was 
thus divided into regions in each of which the amount of displace- 
g 
ment lies between definite limits. The contour w 2 = — encloses the 
y 
regions in which the maximum 
is contained : — and (where 
