732 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
indices of the exponentials of the first fine, and — (x — x) in all the indices of the 
second line. These three pahs of terms will be called W l5 W 2 , W 3 . 
Fortnightly term. 
Tills is f(| 
Multiplying (36) by (28) when the symbols are accented, and only retaining desired 
terms, 
-§(-§- — Z ,2 )(-j—27~) = 1|(-^— 2tn7WKK:) (^ —2rffrrr ffff)-bfHuyhfWX^e ~' 6 d) 2h 
+|HrfA , V (fl - 9)+3h . (39) 
Even if y had been accented in the X' functions, neither y or y' would have 
entered in this expression. These terms will be called W 0 . 
Then the sum of the three expressions (37), (38), and (39), when multiplied by tt'/Q, 
is equal to W, the disturbing function. 
If Diana be a different body from the moon the terms in 9'—9 are periodic, and the 
only part of W, from which secular changes in the moon’s mean distance and 
inclination can arise, are the sidereal semi-diurnal and diurnal terms, viz. : those in 
F and G, and also the term independent of H in (39). These terms being independent 
of 6' are independent of e, the moon’s epoch. Hence it follows that, as far as con¬ 
cerns the influence of Diana’s tides upon the moon, dWjde is zero, and we conclude 
that —the tides raised by any one satellite can produce directly no secular change in the 
mean distance of any other satellite .* 
But Diana being still distinct from the moon, the F-, G-, and part of the fortnightly 
term, which are independent of 9, do involve N and N '; for W contains terms of the 
forms e ±aN , e ±aN ’, e ±{aN+f3N ' } , also it has terms independent of N, N'. Hence dW/dN' 
will contain terms of the form e ±aN ', e ±{aN+ ^ N ' ) , or their equivalent sines or cosines. 
Now by hypothesis there are two disturbing bodies, and we know by lunar theory 
that the direct influence of Diana on the moon is such as to tend to make the nodes 
of the moon’s orbit revolve on the ecliptic ; on the other hand, there is a direct 
influence of the permanent oblateness of the earth on the nodes of the moon’s orbit. 
If the oblateness of the earth be large, the result of the joint influence of these two 
causes may be such as either to make the nodes of the moon’s orbit rotate with a very 
unequal angular velocity, or perform oscillations (possibly large ones) about a mean 
position. If this be the case the mean value of d.W/dN' may differ considerably 
from zero. This case is considered in detail in Part III. of this paper. 
If on the other hand the oblateness be small the nodes of the orbit revolve with a 
* If there be a rigorous relationship between the mean motions of a pair of satellites this may not be 
true. This appears to be (at least very nearly) the case between two pairs of satellites of the planet 
Saturn. 
