218 
SIR G. H. DARWIN ON THE 
Combining these formulae, I find 
A 4 
(7t 2 _ ^1 
Xt3 — -- - 
4 sin 
7 
[{tan 4 y— -§ tan 2 y + 
The form is somewhat awkward, 
better shape. 
Y(n-i)}{i+|^ 2 +^V 4 } 
— k ,2 {\ +- 52 K' 2 ) tan 6 y+yf#c' 4 tan 8 y] . . (60). 
but I have not been able to reduce it to any 
§ 15. The Values of ©f for High ,er Harmonic Terms. 
For higher harmonic terms it is necessary to adopt the approximate forms of the 
functions investigated in “ Harmonics.” The development is there carried out in 
powers of a parameter /3, which I will now write /3 () to avoid confusion; this 
parameter is equal to — ■ - 2 , or to ^/c /2 + |-/f /4 ... of the present paper. 
J- 1 K 
The functions are here defined by 
w = % 
dv 
(v)J{v 2 -lf 2 [v 
1-/V 
but in the notation of § 10 of “Harmonics” this would be called ( 1 >)/%*. Thus, 
if ( y )] denotes that function as defined in “ Harmonics,” we have 
a s (v) 
[€/ fr)l 
We have for the approximate expression for 
Vi M = Pf (v) + A#. W* M + Poq s+ 2 Pi S+2 (v) + fo*q'-*Pr* {?) + A, 2 ? S+ 4 P / +4 (v). 
The investigation on p. 500 of “Harmonics” shows that the leading term of 
[<&/ M] is 
/_ \s ‘‘T t t + .s! 
V ’ 2i+ 1 ! v l+l 
1 + fi 0 q s - 2 
i+s— 2! 
i+s ! 
+ A)<?s + 2 
i+s + 2 ! 
t+s! 
f+s-4! 
i + s ! 
T fiofs+i 
t+g+4! 
i+s ! 
This has to be divided by 5£f, the formula for which is given 111 § 10 of “ Harmonics,” 
and we thus obtain the leading term of a s f). 
For the second term it will suffice if we take /3 0 as zero, so that it is only necessary 
. * ,1s 
to consider Q/(v), which is equal to (v 2 -l) s/2 ~ 
