BOUNDARY PROBLEMS AND DEVELOPMENTS. 



123 



the principal matrix on the right again breaking up into three compo- 

 nent matrices. Of these the first, 



1 / j- HW-TWS^BW-Bimx T . fc (X)4«gCe Xr * W - B * (6) /A(6-0)), 



and the third, 



- 1 



** 









ra(X)w[ a X h f e-^ (l) -ie- B ^%(t)Ut 

 Y dt 



are for all values of x clearly of the type (e)/\. Only the second 

 matrix of the sum, namely 



Ja(x)- = ^ ■( Z e M^)-«,-l^^^^» r tt (X)»iM/ 4 (fl+0) , 



X \A-,ft=l / 



remains, therefore, to be considered. It depends for its character 

 upon the value of x. 



For x d(i a, x =1= b, the matrix is clearly of type (<p2)/X. For x = a, 

 on the other hand, the exponential factor common to the elements of 



the i th row reduces to e -{ xr «W + **< 6) }5« = S* ( + e, so that 



J 3 (a) ■ = — ( £ S *ii T »*( x ) W M 8 Thfh(a + 0) + e) . 

 The fact that 



J 3 (b) . = ZlY £ 5**r a (X) u&> 5*,; A(a + 0) + ^ 2 ) , 



may be established in similar manner by use of the relation 



Substituting for (t,-,) its value as given by formula (115) it is readily 

 seen, therefore, that on arc C M „ 



