168 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
values of @, 6 which define the source, given in (8). For these are ‘ 
g= J, {es-22dueR - eta 
0 
ex@-2) JKR — e-*h sala 
| e 
taogf foment 8} 
4 yee 
eX-2) JKR — en Kh 
See | ner K2-2Z)J Rak, if 2>2 
K 
oo Z| “ere 2)J kRdk, if z<z’, 
K 4 
(60) 
These will obviously be reproduced, after peau by removal of terms of negative — 
degree and integration, from 
p= penne ATR, ifz>2' 
K 
ayes = ex@-2)J eR, ife<z 
K 
aK 
=I = +2! Jone) yeR, if z<z’. ae 
K 5 
When these last terms are taken in, the first lines of ¢, @ in (59) become 
p= be cosh «(z—2’)J xR 
6= a cosh K(z—2')JokR Fz sinh «(z-2’)JI xR, 
the upper or lower sign being taken in the ambiguities according as z> or <2, 
Hence, when the source is taken in, the following are the unprepared and unintegrated — 
forms of @ , 0 :— 
1 
2K 
b= +— cosh x(v—2’)JoxR 
sinh «zJocR { ne ee . = 
aon) dh sh 2xh) cosh 
«(sinh 2Kh — 2xh) xz’ sinh Kz’ — $(a + cosh 2«i) cosh xz j 
wear bo by h) { «Zz cosh xz’ + $(cosh 2«h — a) sinh xz’ j 
6=F 5 cosh x(2—2')S xR FZ sinh x(z — 2’)J KR 
K 
sinh x2J xR PAG iS ies ' 
enh Dei Kz’ cosh 2«h sinh xz’ +( = cosh 2«h + 5g + 2K h *)eosh Kz i 
cosh KzJ )kR ' 3 a a é 1 Sai NE j 
ate dein Dee ED) { kz cosh 2«h cosh xz’ + (- 7 cosh Qh + 3 + 22h ) sin Kz i 
in 0, L/?®+M/«; and these terms, as in 
plate. 
