220 
SIR G. H. DARWIN ON THE 
§ 16. The Fourth Zonal Harmonic Inequality. 
In developing the expressions for the higher harmonic inequalities it seems to 
be most convenient to retain the parameter /3 0 , which is equal to - ~ K ~-. instead of 
developing in powers of k ' 2 as heretofore. 
On putting i = 4 in (63), we find 
k 5 
k 2 
® 1 ( £ j - 5.7 • 9 r 5 1 + It ) ( 1 - 3/3 0 + o 2 ). 
With the notation of “ Harmonics ” we have 
= F i{ v ) + ifioPi(r) + ris A) 2 ^4 4 (v), 
(</>) = l — 5/3 0 cos 2</> + f|-/3o 2 cos 4(f). 
Accordingly 
^ 4 (1) = P 4 (1) = 1, 
C 4 (in) = 1 + 5/3 u + ff/V- 
Again, from § 22 of “ Harmonics” for type EEC, i =4 , s = 0, we have 
j HMEJ’j pda- = (v‘- |i|) 1/2 [l + 10/3. + ^^. 
But is this integral multiplied by 9 and divided by 3 times the volume of the 
ellipsoid. 
Therefore 
=1 + 10A, + -J - /3 0 2 . 
In this formula the coefficients of the powers of /3 0 increase with great rapidity, 
and the approximation may not be very satisfactory; nevertheless it is the best 
attainable without an enormous increase of labour. 
Combining our several results 
‘3m 4 ® 4 ( k ) - 
6 ^ / T i 1 5 
105X r 1 
3 1+m (1- 8 A+^FA 2 ) • (64). 
It remains to find P 4 (^ 0 ), and the denominator in the expression for J\ which 
involves 
We have 
^1 4 = €ti{v 0 ) 
12 f dv 
= 
J -'o®4W] 2 (P-1) 1/2 (P-1/P) 1/2 
Inside the integral sign I write r = — . and change the independent variable 
k sin xf/ 
