PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
273 
+ 2 <{ 
n = 2 
y 
If we now seek solutions for which hx is a multiple of sinh na or cosh not, we find 
the following solutions which are not included above— 
hx = (E cos /3 + F sin /3 + G cosh a+ H sinh a) /3. 
Since any constant multiples of x, y and any constant may be added to x without 
affecting the stresses, it follows from (2) that any multiples of 
sinh a, sin j3 or cosh a—cos (3 
may be added to hx- This allows us to take the coefficients of cosh a, sinh a and sin /3 
as zero. W e have then the following general expression for hx -— 
hx = (E cos f3 + F sin /3 + G cosh a + H sinh a) (5 
+ (B 0 cosh a + D 0 sinh a) a 
+ (A : cosh 2a + B 1 + C 1 sinh 2a cos (3 
+ (A , 1 cosh 2a+ 0^ sinh 2a + D'ja) sin /3 
[A„ cosh (n+l)a + B„ cosh (n — I) a + C„ sinh (n+l)a 
+ D n sinh ( n— l) a] cos n/3 
+ [A' cosh (n+ l) a + B'„ cosh (n— l) a + C' n sinh (w+ l) a 
+ D' n sinh (n— l) a] sin n(3. 
- - - (21) 
We have now to determine whether the displacements corresponding to this stress- 
function are single valued or not. The function (hQ) is easily obtained by simple 
integration from (16), and the arbitrary functions thus appearing can be determined 
by the aid of (17). We have 
— — (hQ) = (E cos (3 + F sin (3 + G cosh a + H sinh a) a 
A + 2/jl 
-(Bo cosh a + D 0 sinh a) (3 
— (A] sinh 2a + C X cosh 2a + D' 1( 8) sin (3 
T (-A/ x sinh 2a + C / 1 cosh 2a — D 1( 8) cos (3 
f [A'„ sinh (n +1) a + B'„ sinh (n — 1) a + C'„ cosh (n +1) a 
I + D' n cosh (n— 1) a] cos n(3 
— [A„ sinh (n +1) a + B„ sinh (n — 1) a + C n cosh (n + 1) a 
+ D„ cosh (n — 1) a] sin n/3. j 
. . . ( 22 ) 
It is clear, from the general expressions for h-x and hQ, that the only terms which 
can possibly give rise to many-valued displacements are 
hx = (E cos {3 + F sin (3 + G cosh a + H sinh a) f3 
+ (B 0 cosh a + D 0 sinh a + D x cos (3 + D'j sin f3) a, 
2 Q 
+ 2 
n = 2 
VOL. CCXXI.—A, 
