10 
1’10JFESS(JK A. E. 11. LOVE ON THE INTEOEATION OF THE 
'N.> 
(y 
dr- 
n(ii + 1)] y _ 
\ dr r 
9 , >i\ / 9 'II, + 1 \ 
3-- + r! ^^[ 3 ; + — 
-■ ") 
and 
■9 
9/- 
rJ \ar 
n — 1 
\v 
liere F i,s any function of r, we find for </> the form 
.j, = ,-s. (I 
(13) 
^\•llere F and /’are arbitrary functions. 
It is now easy to Avrite down simultaneous solutions* of the system of equations 
(8) and (9). Tidving w,, to represent a spherical solid liarmonic of positive degree n, 
and writing 
^0 — ~ c^),.(14), 
a set oi’sucli S(->lutions is given l)y the equation 
(m, r, (r) 
9 9 
^9y’ "9v 
(15) 
and a. second set of such solutions is obtained by taking the curl of the first set. We 
should fnd for example, after a little reduction, 
9/r ^ cv _ 'll + 1 /'I 9 / 1 9-0,A cco„ _ li _+ .S 9 / \ 
9y 9,3' 2/3 + 1 \ ?’ dr/ \ r dr~ J ' d-r 'hi + 1 \ r dr ■ \i‘j' dr \r~'‘ + ^J 
.... ( 1 ( 5 ). 
4'he scdutions given liy (15) may be referred to as “solutions of the frst tyjAe,” 
and those given by equations such as (10), as “solutions of the second ty])e.” 
1 I. For our immediate purpose it Avill l>e suffcient to take a = 1 and W[ = .r. The 
Components of a vect(.)r Avluch yields a solution of the frst t\q)e are 
0, 
/I 
\ r' 9/' 
(i9) 
and the components of the curl of this Amctor are 
- ,ry 
— .r: 
J; d^ _3 9(^1, 
r'i 0^2 ,.t 0/. 
1 9~0u 9^, 
y/i 0/'2 /■' 0/' 
’ 9-(l)„ dd'h 
/■•’ 0/-- /■'• dr 
* 5\'ht'ii the fuiietiuiis that oeeur arc simple harmmiie fimctiuiis of the time, the solution of the 
iirinilihed system of epuatioiis is well known. See L.VAli:, ‘ llyilri/dynamies, j'p. -187 and .jDj r/ stf/. 
