474 
PROFESSOR A. SCHUSTER OH THE DIURXAL 
cos 3?f = 
cos 4 ?a = 
8 ^^^4 3 ^^^2 
dfJi clfj. • • • 
6 4_ _ 16 ^^^3 I_3_ 
r/^ (V 
COS 5u = 
cos Gu = 
12 8 
df. 
24 
” dfi df^ 
5 1 2 
3 0 0 3 
r/P^ 
d/j. 
i2 8. 
4 5 5 
^^5 I_4 ^ ^ ^^3 I 
r//x 
dfU, 
COS 
u = 
df^ 
(C), 
P)> 
(E) , 
(F) ; 
(G) , 
cos 2ii- 
_4_ 
dfjC^ 
7 ^ 
T3A 
(H), 
cos 3?^ = 
8 
3 46 5 
f/;a3 
(I), 
COS 4w = 
cos 5w = 
cos Gu — 
_6 4___ 1 6 ^’^^5 1 7 ^^^3 
45045 1 7 5 5 ^^3 + 29 7 V 
1 3 5 1 3 5 1 48 5 i" 429 ^;^3 
512 _1_ 
7 6 5 7 6 5 ^^^3 3 6 465 ^^^3 "1- 5 8 5 
_7_^ 
1‘^8 7 ^^3 • 
(K), 
(E), 
(M). 
There is another formula useful in similar investigations, which may find a place 
here, although I have not used it in the final reductions. It is the expression of a 
zonal harmonic in terms of the n'"'^ differential coefficient of other zonal harmonics. 
p_ { 2i + 2?z- + 1 )_ 
' ~ (2i + 2n + 1) (24 + 2n-l) . . . {2i + 1) 
n {2i + 2n — 3) 
c/"P- 
^ I + 1 
r/"P' 
'' 1- / + « — 2 
+ 
1 (24 + 2a - 1) (24 + 2a - 3) . . . (24 - ]) " ^ + 
a.a —1 (24 + 2?i—7) 
1.2 (24 + 2?!, — 3)... (2?; — 3) 
24-(2a-l) 
cPP,- 
« + « — 4 
fPP 
*- i — n> 
(2/ + l)(2(-l)...(24-(2a-l) 
the general y)*” term being 
, , , a.74 — 1... a — (» — 2) 24 + 2a + 5—4» 
'' ' 1.2... a-1 (24 
(24 + 2n + 3 — 2p) (24 + 2?i. + 1 — 2p)... (24 + 3 — 2p) 
ff“P 
tt X — 
i+n—2p+4* 
The proof is conducted exactly in the same way as for equation (3). If i is equal to 
or greater than 2n, the series has its full number of terms, viz., n + 1, otherwise the 
series breaks off owing to the differential coefficients vanishing. 
