199
there are regions, such as those surrounding the Poles, providing
special problems and perhaps unexpected obstacles. I shall be
surprised if the whole surface is entirely and adequately surveyed
within 50 years. It behoves us then to seek an escape from this
position.
For the present section, I will write my formula in the approxi
mate form, in which it reduces to that of Stokes: thus
NG Ag dco
with which you will be very familiar.
In this integral the function is of vital import. Let us consider
its nature
f S T +r Pn 5 p2 3-5 P3 3^4 2-75 P5 2.6P6 2.5 P,
etc.
The function of each P is to give due effect to the corresponding
harmonic constituents of Ag. If such constituent is zero that P
term could be omitted, for it has no effect on the other harmonics
in the complete integral. Use of a portion only of the integral is
inadmissible (though I am not sure that this has always been
observed!).
The early harmonics are taken into account in the formula for y,
so that only a small residue of the second harmonic should remain
in Ag.and the part associated with cos 2X or sin 2X.
I can only suppose that the third harmonic, for which no dynamic
cause is known to me, is likely to be small: and the same for the
next few harmonics. Artificial Satellite study has indicated that the
coefficients of the zonal harmonics (with reference to the axis of
rotation) of order 4 and 6 are smallbut unfortunately the probable
error of the determination is considerable.
I am proposing to modify the coefficients of the Legen dre
coefficients in up to order 7,and to form a quantity A 2 cm P>n
chosing the coefficients c, so that r and Ahave the same values at 8
values of
You will understand my purpose and see its effect, if you will
look at the equations (7) to (9) which I have distributed: and in
figure 1 the situation is shown graphically.
I propose to use A/ in place of in the integral. That would
make no difference if Ag contained no constituents of orders o to 7
and in (9) I have indicated the effect if these components have the
values which were given in A.M.S. Technical Report No. 24 ^Wash
ington) of 1959. I do not think that those values, derived from the
present available data, are literally truebut perhaps the}' at least
2 rv i
7
0
