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SV +0"001 C sin {+(S-T)+j'S+ a}, Saturn.

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3. In order that the discussion of the various differences may be made clear, it is necessary to state certain facts in the theory of these inequalities which have considerable effect on the final results, and on the methods of obtaining them.

The disturbing function R is of the form

R={A, cos Q+A1 cos (l+Q) + A, cos (2D-1+Q)

+ A2 cos (2D+1+Q) + A ̧ cos (2D+Q) +A5 cos (2D - 21 +Q) +

},

where Q here and throughout this discussion is an angle depending only on the solar and planetary arguments, and where the summation sign refers to the different arguments Q.

Let w1 be the Moon's mean longitude, and c any one of the other five lunar elements. When the value of R has been substituted in the equations of variations and the latter solved we obtain

Sw1 = {B sin Q+ B1 sin (l + Q) +

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the term in dc being a sine or cosine according as c is or is not an angular element.

The values of the variations of the elements have to be substituted in—

say,

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Now V consists of a non-periodic term w, and periodic terms,

V=2+v1 sin l+v2 sin (2D − 1) + vg sin (2D) + 1) + ....,

-

and from this we obtain easily the values of the deviatives of V. The terms which arise from the substitution of the variations of the elements in 8 may therefore be divided into two classes. The first of these, which I have called elsewhere* the primary terms, consists of those terms in 8V which arise from the variation of the non-periodic term; the secondary terms are those arising from the substitution in the remaining, that is, the periodic terms of V. Thus the values of dw, constitute the primary terms, and all the other portions of 8V the secondary terms.

Now the primary terms have the same arguments as the terms in R, that is, the arguments

Q,1+Q, 2D-1+ Q, . . . .,

while the secondary terms have the arguments Q±(/), Q±(2D-1), Q±(2D+1),

7 + Q ±(1), 7 + Q ±(2D-1), 7+ Q±(2D+1),

* The Inequalities in the motion of the moon due to the direct action of the planets, Pitt Press, Cambridge, 1907.

the arguments arising from the derivatives of V being enclosed in brackets. It is therefore obvious that the arguments Q, /+Q, . which are present in the primary terms will also appear amongst the secondary terms; they are due to the separation of a product of a sine and a cosine into the sum or difference of two sines.

4. The first result to be noted is the fact that the sum of the secondaries with arguments l+Q- (), 2D - 1 + Q-(2D-1), that is, those having the argument Q, is very small compared with the coefficient of the primary with argument Q.* In many cases these secondaries are of the same order of magnitude as the primary of the same argument, the chief of them in general being those arising from the two arguments just written down. This theorem is a consequence of the method of the variation of arbitrary constants. If we had adopted the straightforward method of integrating the original equations for the Moon's motion with the additional value of R due to planetary action, the terms independent of in the coordinates arising from the terms in R which contain l, and which therefore have the factor e (the lunar eccentricity), would have possessed the factor e2, while the principal terms due to RA, cos Q would not have this factor; the former terms must therefore be quite small.

This fact constitutes a useful test of the general accuracy of a large portion of the work, and it was satisfied in all cases as closely as could be expected. It is illustrated in the following table for a few terms of the indirect action of Venus in longitude. The first column gives the arguments of the primaries (that is, of the terms in R), and in line with them are the resulting secondaries for the values of which stand at the head of each column. The last line but one gives the sums of these secondaries, and the last line the coefficients of the primaries. The sums of these two lines constitute the complete coefficients for the indirect action. The constant angle to be added to each argument is omitted, as it is (in these and in most cases) very nearly the same for each of the numbers in a given column.

SV=0"001C sin (+ a), iTV, Venus. Values of C.

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* This result was not noticed until the computations had been practically completed, when the approximate vanishing of the sums of the secondaries in all cases pointed to a general theorem.

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