4 
SIR G. H. DARWIN: FURTHER CONSIDERATION OF THE 
We know that ©/ (</>) (the cosine function of <f>) is the same function of — k 2 sin 2 (j> 
that (/x) is of k 2 cos 2 6, except as regards a constant factor. 
Hence it follows that 
(<£) = X 
a+b^j sin 2 (f> + c ^ sin 4 (f) + cl~ sin 6 <f>+.... 
K. K. K. 
where A is a constant factor. 
Now I desire to define (/x) and (£/(</>) exactly as in “ Harmonics.” 
This definition has already been adopted as regards (p), but it remains to adjust 
the constant A so as to attain the same end as regards C/ (</>). 
When i and s are even, C/ ((f)) was defined thus : 
= cos S(f)+(3p s+2 cos (.s+2) (f> + i3y s +i cos (s + 4) <£ + ... +/3 }(1 s) pi cos i(f> 
+Pp s - 2 cos (s—2 ) (f> + /3y s - 1 cos (s— 4) (f> + ... +f3 hs p 0 . 
Since 
sin2r <l> = 
27 -! 
2r ! 
2 2r (r !) 2 2 2r_1 (r—1)! (r+1)! C ° S ~^ + 2 2 ''" 1 (r-2 )! (r+2 )! 
it follows that the term independent of (f> in C» ((f)) is 
2r ! 
cos 4(f) —, 
, 17 /c' 2 ,1.3 K n , 1.3.5 7 k /6 , 
a + ±b— + —-c — +-——CI — + 
k 2.4 k 2.4.6 k 
The term in cos 2 (f> in (£/ ((f>) is 
■2A cos 2 (f> 
T 1 7 k ' 2 
b% + 
K 
1.3 2 k' 4 ,1.3.5 3 7 k' 6 , 
- . — C —7 + - . — cl — -r + 
2.4 3 k 4 2.4.6 4 K b 
The term in cos 4(f) in Cd (< f >) is 
2A cos 4<f) 
1.3 1.2 K n 1.3.5 
2.4'3.4 ° k 4 + 2.4.6 
2.3 7 k /6 , 1.3.5.7 3.4 ac /8 , 
- a — H--- e — Q + 
4.5 k 6 2.4.6.8 5.6 k s 
and so forth. 
In accordance with the definition to be adopted, these terms in the three cases 
respectively are : 1, cos 2 (f>, cos 4 (f>. Hence A must be chosen so as to fulfil that 
condition. 
Pursuing only the case of (£,• (<£) in detail, we have 
l b — 
2 V 
+ 
1.3 
2.4 
c 
K 
n 
K 
+ 
• • • 
