Scientia Sinica

4 SCIENTIA SINICA Vol, V

When Z is a polynomial in x, y, and 2, a special solution of F can be easily obtained, since it may be taken as a polynomial. When

Z = Acos ax cos By cos yz , (12) a special solution of F may be taken as

By3 A cos ax cos By cos yz

‘a By, By (a? +B? Fy; y?)(@ +B? +02 7) ’ (13) where il 1 s ee eae (14)

_ In general, a special solution of F may be obtained by Fourier series or integral. This method has been used by G. F. Carrier"! However, triple Fourier integrals are usually not convenient for calculation. -

In his paper [11], the author has proposed a simple method for obtaining a special solution. Let

O?F PRE = Gl (Xin a2) VA Bw ane — GeV a12)) (5) ST Oz

i 2 oe ViM la a:

Substituting these expressions into equations (10), we get

OG; , 0G; 1 0G, iB See Ox? Oy? ap x? Ox? az Babu Z (a); Wz), (¢=1,2). (16)

Since we are interested in a special solution, we may consider the body to be an infinite space. In this case, the solution of equation (16) is

G; Gaye) Tah

— By é Si {I Z(E, 7, €) d& dy dé : By By 4m JN) @=a) Gay) bsCoeye ©

=1,2). (17)

After having determined G;, F may be obtained by integrating equations 2 (15). Regarding viF and om

Oz?

solving system (15), we get

as two independent algebraic unknowns and

1 Wii = G,— 2 G 3 1 yj — (v; G; — v2 G2)

(18) OF 1 Oz? Vi— D2

(G; — G)).