Parameter Values of Potential Tlieory : 1 69 



Regarded, however, as a function of p, G(qp) is a double stratum 

 potential of moment XG{gB), together with potentials g{qp) and 

 — Q(gp). From the boundary properties of these we deduce 



(16) ^G{gt + )={1 + \)G{gt) - XMi^t) 

 lG{qt ) = {l-\)G{qt)+XMqt) 



Finally H{sp) regarded as a function of p is the sum of potentials 

 h{sp), —'P(sp)IXg, and a double stratum of moment X/I(s$). From 

 which it follows, in virtue of (14) that 



(17) I H{st + ) = ( 1 + X)II{ St) - V{st) 

 il{{st-) = ( 1 -X)II(.st) + P{st) 



§3. — Solution regular at a singular parameter value. — We are 

 now in a position to find solutions to the boundary problems (1), 

 with second members modified, having no singularities for the 

 characteristic number X^. If we define the functions W{p) and 

 V{p) Ijy 



(18) ^w{p)=/i{d)H{ep)de 

 lv{p)=fG{pd)i{e)de 



we find on suljstituting the values of H{Op) from (8) that W{p) is 

 the sum of potentials of double strata of moments f(;;), -/"f (^)P^0^^5 

 and X/'i{6)JI{6t)d6 respectively. 

 Hence we find that 



h[W{t+) - ir(^-)] -u[ir(^+)+ w{t^)] 



= i(t) -/i{e}F{et)dd + xfi {6)H{et) do 



- xf{ f {(f>)h{(f>t) -ft {d)F(e(f>)h{cf>t)de + xfi{e)H{0(fi)h{(jit)de]dcf> 



In virtue of (8) the second member disappears except for the first 

 two terms. So that W(p) satisfies the boundary condition. 



(19a) i,[Wit+)- n'{t-)] - hx[ W{t+) + w{t-)]=f{t) -fi(e)F{dt)dd 



In this all the function.s are regular when A=/\„ ; so that 

 this equation admits the solution W{p) which is regular even 

 when X is put e(|ual to the singular value A^. It has been shown 

 elsewhere^ tliat for this value of the parameter tlie first problem (1) 

 does not admit a solution by double stratum unless the condition 



/f(^)P(^0^^ = 

 is satisfied, in whicli case the solution is obviously TI (/?). 



Similarly substituting the value of G{pd) given by (8) we find 

 that V{p) is the sum of potentials of simple sti'ata of densities i{t), 

 -j"P{te)i{e)d6 and X/H(te)i (0)dO. From the boundary pro- 

 perties of simple strata it follows that 



= i{t) -f P{td)i{e)de-\-xfH{te) f {e)de 



- xf/i{tcf>) [f (<^) -/'P{ci>e)t(6)dd+x/Ji(<fiO)i{0)dO]d<f> 



1 Weathevburn. " Boundary Problems, etc," J (J. 



