DIFFERENTIAL EQUATIONS 
355 
The first coefficient in (2), namely, a Q , is arbitrary. From (4) we 
find : 
1 
a 2 — — a °’ 
Furthermore, since a x = 0, it follows from (4) that a 3 = 0, ct 5 = 0, etc. 
Each a n is thus seen to contain a 0 as a factor, and since we do not 
care particularly what constant factor is multiplied into a series (2) 
which yields a solution, we will set a 0 = 1. (2) then becomes the 
series which defines the function known as J 0 (x): 
j 0 (x) = 1 - 
(5) 
This series converges for all values of x, and it converges rapidly. 
The series for J n (x) and K n (x) will be found in Peirce’s Table of 
Integrals, p. 87. They can be verified by direct substitution in the 
above differential equation (a). When n is not an integer, J n (x) 
and «/_„(») afford two linearly independent solutions of (a), and there 
is no need of introducing a function K n (x) (which in this case is not 
defined). Put if n is an integer, J n {x) and J- n (x) become linearly 
dependent, and K n (x) is needed to furnish a second solution. 
Zonal Harmonics. Legendre’s Equation, (b), can be treated in a 
precisely similar manner. If a solution is assumed in the form of 
a power series, 
y = a 0 + a x x + a 2 æ 2 + •••, 
it is found that the relation 
(n + l)[(» + 2) a n+2 — naf] + m(m + iK = o, 
or 
holds for n = 0, 1, 2, •••. The coefficients a 0 and a x are arbitrary, 
and we get two linearly independent solutions by setting first one 
of these coefficients, and then the other, equal to zero. 
When m is a positive integer, or zero, one of these solutions re 
duces to a polynomial. For the coefficients a m+2 , a m+4 , ••• are seen 
to vanish, and thus one of the solutions breaks off with the term 
a m x m . The other solution is not a polynomial. 
Let the polynomial solution be arranged according to descending 
powers of x : 
a m x m + a m . 2 x m ~ 2 + ....