5 Sir Thomas Havelock 
with the notation 
1 ‘ 1 
A =| (tan? + ¢?)—2 effeos? dt, B =|| (tan? O + t2)2 efteos? de. (25) 
0 0 
The integrals A and B can readily be computed, either by quadrature or by expansion 
in powers of £. For 6 = 0, L reduces to 1+ 6 — fe? Hip). 
For any given /, values of L were computed from (24) at intervals of 18° from 
0 to 90°; then by numerical or graphical interpolation intermediate values were 
obtained. These were then used to compute the quantities from (14) by numerical 
quadrature. Expressions suitable for small values of # can be obtained. We find 
from (24) 
L=1+£+/(1—cos@) + 63(4 cos? 9 — 3 sin? 6 + 2 sin 6 cos @) 
— (f? cos 6 — f3 cos? @) log {t/y(1 + cos 6)} — 6? sin? @ log ($y sin@)+..., (26) 
with Iny = 0-57712. Using this in (14) we obtain expansions for the coefficients Z,,. 
The coefficients M,, were computed either by quadrature or from a power series 
m 
in £ which can be found from the expansion 
ie) n—1 
My == & (= 19" | Pag) (BP an als) + 2m-+1) Peale} ye. (27) 
Returning to (15) and (16), once the L and M coefficients have been calculated, the 
equations are in suitable form for approximate solution to any required degree of 
accuracy, at least for moderate values of £. 
Accurate computation has not been attempted, but a somewhat crude approxima- 
tion is sufficient to bring out the general character of the results. Calculations were 
carried out for £ = 0-1, 0-2, 0-4, 0-6, 0-8, 1-0, 2-0 and 3-0. As an example of the 
numerical values, we find for = 0-4, 
Ig = 1-4707, L, = 0-2391, L,=—0-0582, L, = —0-0547, 
M, = 0-2464, M, = 0-1400, M, =—0-0428, M, = 0-0262. 
With these values we solve the first four equations from (15) and from (16) for 
eight unknowns, neglecting the unknowns of higher order; this gives 
C= 0-3029, D= —0-0486, A,=0:2012, A, = —0-0352, 
A; = 0-0193, B,=—0-0146, B,=0-0039, B, = —0-0027. 
These may be compared with the corresponding values for the limiting case 
£ = 0, namely, 
O=0:5, D=0, A,=0-2083, A,=—0-0375, A,=0-0145, A,,=0, B,=0. 
6. The resultant hydrodynamic pressure on the sphere is given by 
oF 
Z=- 2mpa? | sind cos bdé. (28) 
606 
