putting A2a+a+, B=2b-b'+s, C=2c+b'+o, D=2d+c'

1175. Substitute this value of P in the general equation < 1110. and (dividing by p) we get

m causes the greatest error, on account of the divisors 1

d

=0,0838, and

× a = 0,9917 x 0,005595 = 0,00555; therefore 1 m2 = P 0,008388, consequently 1-m = 0,004186. According to this conclusion therefore, the mean motion of the apogee: that of the moon:: 0,004186: 1, whereas the ratio ought to be that of 0,008455: 1; this conclusion therefore gives the motion of the apogee only about one half of what it ought to be according to observation. We must therefore see whether we have not omitted any terms, which, in the present case, may have been too considerable to be neglected.

3r2j 2d 3

x sin. 2z, and into

× cos. 2z, we assumed it equal to 1-w cos. mv (1166); whereas, if we

have a very considerable effect in the value of

2

n

- m.v+&c. will be found to

g, or of E, and consequently of

1-m. Now that term which will produce any considerable effect on E is

20

in the values of sin. 2z, cos. 2z. We will take each case separately.

two first terms we have already considered; we have therefore to assume

20

-)

d+

-m. v, and taking only the first term, sin. in the value of sin.

n

, which last term we omit, as not being of the species

3ar3
2d3

At; hence, we assume cos. 2x aq' cos. my, which is

cos. 2% in the value of P.

sit assuming d= 1, in the quantity we introduce the term sq

m. v; and taking only the first term, sin.

-- therefore

rr 2

n

- m. v; hence, we assume

2v

in

n

n

mit, as not being of the species we here want; hence, we assume

::ss. The three corrections therefore which P receives make - (6d

m) aq' cos. mv; but in the value of P before, this term was—a cos.

w; and we put E='; we must now therefore put E=}6+

2

n

2

—m) q'; E is therefore increased by (6 +-4× '— m) q' = 0,0784; and

Ead

3

pm

n

the former value of being 0,0838, the corrected value of E=0,1622; hence, 1-m2 = =0,016236, and 1 m = 0,00836, which is very near 15 rep its true value 0,008455. Now there are, after this correction, only some very small quantities omitted; the operation therefore ought to be a very near approximation, and accordingly we find it to be 'so; hence, we may conclude, that the theory of gravity is sufficient to give the true motion of the moon's 10 godet one 0919 apogee.

1189. Having determined the value of m, we might correct all the quantities which depend upon it; and then proceed to find the correction of the time, and thence that of the true longitude, in like manner as we obtained the same for the planets; taking into consideration the excentricity of the earth's orbit, and the inclination of the moon's orbit. And in the same manner as for the moon, we may find the motion of the apogee of the orbits of the planets. But a full investigation of all these things would exceed the bounds to which we must confine this work.

-m.vx, which last term we may omit, it not being

n

}

correction therefore of - 2e in the value of P is - 6adq COS. ??

we get the term s

which has not been considered; and taking only the first te.

mv + & q cos. -m. v, which last term we omit, as not

4 n