54 
MR. F. W. DTSON ON THE POTENTIAL OF AN ANCHOR RINO. 
dz 
dz 
(D) 
~ I sill x + i sill 2 x 5 4 X + 3 % sin 3x^ 3 -“ 
"*■ ^ 
, ( m + 9 . 360 Z + 123 . ^ , ,, • . \ , , 1 
+ -sm 3x 4 2^8 sm ox j 4 . • .| 
1 
4 
H) ^ + (i 2 x + 
36 / + 31 
sill 2 x 4 (I sin x + i sii^' 
#2 sin 
4x) 
sill X 4 its sin 3x 4 ih sin ox) 4 ■ 
f/J 
2 ' 
dz ~ ” cHF 2 x 4 i sill 4x) 3 
4 (i4 sin X + A? sin 3x 4 W 2 sin 5x) s~ 4 • • •] 
= “ ^TjTi [sill 4x 4 (i sill 3x 4 i; sin 5x) *• 4 • • •} 
'^'^5 , • r I ) 
~ch=^ ~ cHO '^X + ^- -H 
The following forniiike are useful for questions involving the stream-line function. 
They are obtained by multiplying the expansions just given for Jn J., &c., by oT or 
c (I — s cos x)- 
(E) 
^-^ tios X 6- 4 -cos 2x 1 6- 4 ^ ~ 192“ j 
/ 12 / + 11 , 12 /+ 17 . 70 , -0 , ^ . 
—w. _ ^ cos4x)6-^4 
[ 2048 
15/-8 
+ ■'70ir‘““ - a„7: 
JoTTT = 
- j COS X + - i cos 2 x) 6 - 4 i - 
20 /+ 17 , , 
COS X ~ 32 ‘^CS OX I i>~ 
32 
, / 20 / + 47 8 / + 1 ^ 
4 (— 7 R 7 -“ COS 2 x — rig cos 4 x •s'^ 
128 
64 
/ IS /+ 29 11/+1 \ r . , 
+ ^ ~ 256 ■ cos X - ^ - cos ox ~ 2liTs cos 5 x ) 4 4 
256 
•^3^ ~ cHl- i ^ ~ "I ^x) + 
/ 52 / - 57 
32 
f cos 2 x ~ 3^2 cos 4x) 
4 
124 / + 101 
cos X — M- cos 3x — xis cos 5x) 5® 4 • • • | 
64 
