An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics with applications to problems in mathematical physics. By William Elwood Byerly.

138 MISCELLANEOUS PROBLEMS.
log b
2. - m7rrx
where am = X(a e) sin dx.
log b- log a ) log b - log a
0
15: If V —= (r) when -=/, V =0 when 0=O, V =0 when r-a,
and V=O when r-b, then if a<r<b and O < </,
s n- m7rr
'-=z{am log b - log a 7(sin (log - log a)}
V= - - -am N- ^ -sin l b-o a
msrn logb -loga
ml=i sinh
log b - log a
log
2 m7rx
where am = 2 f(aex) sin dx.
m log b-log a- log b-log a
II. Potential Function in Space.
1. Show that
f(x, y) = +J3da jdfl jdxJf(X, ) cos a(A - x) cos ( - y).d,
0 0 -c -oo
for all values of x ana y.
2. Find particular solutions of D 2 V-+, 2D F1+ D-2 = 0 in the forms
V= e- z 2+ 2 cos (aXe ~ y)
V-:e $V/fa2~ 2 sin (ax ~ /y)
V= sinh \a2 + /2. sin (ax ~ fy)
V= cosh z Va2 + /2. sin (ax ~ fy)
&c.
3. Given D2V+ DY,217+-D2V=0, and V=f(x,y) when z=0, solve for
positive values of z.
00 00
Result: V1= I-fd f(X, 7 ---d
27r J[z-+ (X -- x)2 +-( - y)2]
_ 00 00
4. Confirm the result of the last example by showing that if f(x, y) is independent of y
Y= 1 rzf(, gx)dX (v. Ex. 3 Art. 45).
J 2 +(x - x)2
— o0