168 C. E. Weatherburn : 



^2. — Bouiuhtnj discqut inuifitx. — The second of equations (9) 

 shows that Q {qj)). regarded as a function of r/, is the potential 

 of a simple stratum of density ^(Op). From the boundary pro- 

 perties of such it follows that 



Adding and subtracting we find for the normal derivative of 

 Q{qp) on either side of the boundary 



1ao|q(^» = (1-A,)P(^^.) 



Regarded, however, as a fuiictiou of p, Q (qp) is a double stratum 

 potential of moment X^Qi?^)- Henco 

 ) i[Q(^^+)-Q(g<-)] = A„Q(<?0 



» lSQ{qt-) + Q{qi+)] = xjQ{qe)h{et)dd=q(qt) 



Adding and subtracting we have for the values of Q (qp) on either 

 side of the boundary 



(13) jQ{qt + ):={i+K)Q{qt) 



\ Q{qt-) = {l-K)Q{^f) 



Similarly P (sp) as a function of p is a double stratum of 

 moment X(,^{sO) ; and its values on either side of the boundary are- 

 found in the same way to be 



(14) I F{st+) = {]+X,)-P{st) 

 \-p{.sf.-) = {\-X,)P{sl) 



From the second of equations (10) G(qp), regarded as a function 

 of ^ is the sum of potentials g(qp), —Q,{qp), and a simple stratum 

 of density Xff(Op). From the behaviour of tliese at the boundary, 

 and in virtue ..f (12), it follows 



ir^,a^f~p)+ lG(t^p)'] = X/HW)H{Op)dO + /,{tp)-T{tp)/X, 

 Lan ail J 



^H{tp) 

 Addiiiir and sul)sti-actin<'- we find 



(15) 



\^!^ 



{t~r) = {\+x)iJ{tp)-V(fp) 



^a{f+p) = {l-X)H(tp) + F{tp) 



