The following integral relations are needed 
[2 Zo(erae=2 2,00) 
| Z, (x) de =- Z, (z) 
ar Z, (az) Zo (82) - Bx Z, (Br) Z (az) 
|. Z, (ax) Z,(B x) dx = ra 
a safe 
[- be! (az) dz = _ [Z” (az) + Z}(az)] 
When the boundary conditions are applied the solutions of some of the differential equa- 
tions of this report are the cylinder functions Cik (a, y) and the modified cylinder functions 
Dix (a, y). Jaeger* defines these functions as 
Coo (ts ¥) = Iq (a) Yq (y) - Yq (z) Jy (y) 
j+k 
C., (2, y) = : C, (a, 
pW alae? OO y) 
Doo (ty) = Ig (t) Kg (y) - Kg (2) I (y) 
j+k 
Dix (t, y) = el ay Doo (ty) 
These two sets of functions are connected by the relation 
gitk D jy, ia, ty) = - 5 Ci, (ty) 
The modified cylinder functions have the following properties 
1 
Dj krg\tsy) = Dix (x,y) - ; Di ry 2) 
iL 
Diy (x, z) = =D (x, 2) = : 
46 
