PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
271 
and we have still to satisfy (9). Substituting for u, v in terms of P we have 
dot \]f d/3 
which may be re-arranged thus— 
+ h 2 8 ( X ^ + 2/ul v2 
0/3 W da) X + fji. 
X + 2fj. 0y _ 1 oP| _£_ (\ + 2/u d X 1 rP 
0a[X + /u da If d/3 j d/3lX + n d/3 If da 
It follows that a function Q exists such that 
SQ _ 0P _ X + 2/U. jy 2 ex 
ca 0a X + /x d/3 
0Q _ _ 0P X + 2/U J 2 £X 
0/3 0/3 X + /U ca 
= 0. 
(14) 
(15) 
Eliminating P by differentiating with regard to j3 and a respectively, and adding, 
we have 
( /; Q) _ q c 2 h _ X + 2m J1 _0_ i , 2 dx , _ 1 _£_ /, 2 0x \ 1 
0a 0/3 2(X + yu) // 0a 0a h d/3 0/3/ j 
0a 0/d 
which becomes in our co-ordinates 
0MAQ) _ X + 2^ J 0 2 (A x ) 0 ^(/?x) _ r I 
du d/3 2(X + n) [ 0a J 0/3 J 
(16) 
There is, however, a further condition to be satisfied by Q corresponding to the 
condition V 2 P = 0. Differentiating (14) and (15) with regard to a, (3 respectively, 
and subtracting we have 
These two equations connecting Q and x are consistent, for, if we eliminate Q by 
appropriate differential operators, we have 
(h x ) = - 4 
a 4 (hx) 
da 2 0/3 2 ’ 
which is readily seen to be identical with the condition V l x = 0, as given in (8). It 
is obvious that hQ satisfies the same differential equation, and hence it also is a 
solution of V 4 Q = 0. 
