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5. At first sight it would appear that the arguments of the two terms formed, for example, from sums of products like sin (+Q) cos (2D), that is, 1+Q+(2D-1), would have coefficients in SV of the same order of magnitude. It is not so whenever the
arguments in R and V contain 7, unless the argument in R is of very long period (greater than a score or so of years). This fact might have been predicted from considerations of the same nature as those in the previous paragraph. For example, an argument
27+Q+ (1) Q+37 would not be expected from a direct method of treatment to have a coefficient nearly so large as that of 2/ + Q Q+1. The latter is frequently sensible, while there are no terms with the former argument.
This arises, again, from the peculiarities of the method of the variation of arbitrary constants. In the volume referred to above I have shown directly from the method that if a term of R contains il in its argument (i = 1, 2, 3, ...) and V a term i'l(i'′ = 1, 2, 3, ...), then the secondary arising from the sum of these arguments is always very small compared with that arising from the difference, unless the period of the primary is very long. This result was proved by inserting the terms due to Sl, de, as given by the equations of variations, in SV. It enabled me to abbreviate the computations by some twenty or more per cent., since it rendered the separate computation of de unnecessary except in a few easily recognised cases.
6. One other fact should be noted. Although the coefficient of the evection (arg. 2D) is of order m(1/13) compared with that of the principal elliptic term (arg. 7), the numerical factors which it contains are so large that in certain functions of the lunar co-ordinates it becomes equal to or of more importance than the principal elliptic term. It is therefore never safe to neglect the former terms unless we know that the latter are quite insensible. This shows most strongly in the arguments + in R for the indirect action. The terms which multiply SV' (the portion of the Earth's longitude due to the action of the planets) are chiefly those due to the evection, while those which multiply dr' (the planetary portion of the Earth's radius vector) are due to the evection and the principal elliptic term. Yet the former portions in several cases are larger than the latter.
7. The differences B-R. Each of Radau's coefficients depending on the action of Venus is to be diminished by 1/51 of its amount on account of the difference between the value for the mass of Venus adopted by him and that adopted by Newcomb and myself. Each coefficient due to Mercury is to be diminished by 7/60 of its amount for my adopted value of the mass of that planet.
The equations of variations used by Radau appear to require a correction which would increase each coefficient in the ratio 96 95, as I have shown in the volume previously referred to. This is confirmed by Newcomb's results, for he obtains values for the principal terms the same as those which I found in the volume quoted.
In the primary terms with arguments independent of the lunar angles, Radau omits the portions due to the term of R which depends on r2 cos 2(VT). These portions constitute about oneeighth of the principal portions for the indirect action, and his coefficients for this portion should therefore be diminished in the ratio 78. The corresponding correction for the direct action is smaller, and its amount depends on the particular term under consideration.
In the short period terms of the disturbing function for the indirect action of the form + he omits the portions multiplied by SV. These, as stated in No. 6 above, constitute a considerable fraction of the whole. The terms proportionately most affected are' those with this argument, but the differences for the terms of argument + (1) are much larger, for the latter (secondary) terms are about three times the former (primary) terms.
The discovery of the causes of divergence between the results of Radau and myself was rendered difficult owing to the fact that, although his secondary inequalities for the Venus terms of argument
are given separately, and are in several cases quite large when the theorem of No. 4 above shows that they ought to be quite small, yet his final coefficients very nearly agreed with mine. As a matter of fact, I have found, from an examination of the separate portions of my results, that these corrections very nearly balance one another in the case of Venus, though they do not do so with the other planets, nor do they in the case of Venus with arguments other than .
I have compared most of Radau's individual results with mine, and find a close agreement for nearly all those portions which he has computed. For the direct action this agreement was particularly useful, since my coefficients are obtained from formulæ and methods radically different from his, though reducible to them by algebraical processes.
Two other differences should be noticed, namely, those in the terms with arguments - D+4T-3V and 1-D+4T-3V. These arise from the fact that the indirect action is a little greater and of the opposite sign to the direct action; Radau only takes the latter into account.
8. The differences B-N. These have been marked with the letters E1, E2, Eg (explained), and U (unexplained), whenever they exceed o"020.
Those marked E, are due to the fact that Newcomb substitutes the variations of the elements in the value of V limited to the three terms
V=w1 + 2e sin l + že2 sin 27,
instead of taking the complete value; he adds that "in nearly or quite all cases we may drop terms of the second order in e."
This third term of V in reality only produces a few secondaries with coefficients greater than o"020, and these arise in the form 2D-21++(27) = 2D + ; but there are many greater than
o"003, the adopted limit below which coefficients were dropped. The latter arise also in the forms + − (27) and 2D −1 +&+(27).
A much more extended and larger set of differences arises, however, from the neglect of the evection in the expression for V. The majority of the differences E, due to this cause will be found in the combinations
1++ (2D-1) = 2D+, 2D-1-(2D) = 4,
2l - 2D++ (2D − 1) = 1 + $,
the last being of importance only when 2l-2D+ is of long period of the order of ten years, and the first quite rarely. Hence the final coefficients of terms with arguments , 2D+ will be chiefly affected, and a very few in those with arguments 7+, from this cause. All those differences which may be nearly or altogether explained by the neglect of this and other terms in V are marked E1. I assume that the terms not present in Newcomb's list have not been computed.
In order to examine the differences E, a closer inspection is necessary. On referring to Newcomb's tables for the variations of the elements (p. 154 of his work), I find that under the argument 2D he has only the terms containing (VT) and 2(V − T) for Venus. There are, however, several other arguments with coefficients of the same order of magnitude as these, and the same fact appears to a smaller extent with the other planets. All differences explained by this cause have been marked E.
The small difference E, in the great long period term due to Venus scarcely needs explanation. Newcomb has included the portion (less than o"10) due to the mutual perturbations of Venus and the Earth, and he states that his method indicates a possible error of the order o"10. I exclude the former part, but the maximum error by my method should be less than o′′ 05.
Of the differences U there are two in Mars, eight in Jupiter, and one in Saturn. The most important is that in Jupiter with argument J; I have compared the several portions of any value. for the primary with that of Radau, and the results (after the corrections noted in No. 7 above have been made) agree for the indirect action, but I have been unable to obtain his result for the direct action from the algebraical formulæ which he gives. The causes for the remaining differences I have not been able to trace, and must leave the values of those coefficients an open question.
9. In working out the terms produced by planetary action, the valuable memoir of M. Radau has been available for comparison of results at almost every stage, and has materially assisted in the prevention of errors made during the course of a piece of work of great complexity, though not of great difficulty when once the theory had been put into final shape. The work of Professor Newcomb only appeared when I had finished the greater part of
the computations, and would in any case not have been available for detailed comparison without much labour, owing to the complete difference between his method and mine. He combines the direct
and indirect actions at the earliest opportunity. I have kept them separate until each was fully completed. The comparison between his results and mine revealed one error in my work which affected the primaries due to the terms with argument in R by about 5 per cent., and a few of the small secondaries from these terms about twenty per cent. ; an error in the equations of variations had almost no effect. These errors have, of course, been corrected in the results given above.
10. Like Professor Newcomb, I have also made an examination of the inequalities of the second order relative to the planetary masses, and have so far found nothing that could sensibly affect the motion of the Moon. I have found an additional portion to the term with argument + 3T - 10V of the order of o":2, but the period of this term is so long that it would scarcely affect the observations within the degree of accuracy at present obtainable. The motion of the node of Venus also affects the term with argument +16T - 18V by a quantity of the same order of magnitude. The terms of the second order in the Sun's motion, as given in Newcomb's Tables of the Sun,
¿V' = −0′′ 265 cos (4M1 - 7T1 + 3V1)-o"021 sin (4M, − 7T1 +3V1) +376 cos (3J1-8M1+4T1)+5'18 sin (3J1- 8M, +4T1) (where the suffix denotes the mean anomalies of the planets instead of the mean motions), or
+0" 266 sin (1°190t+31°8)
(period 300 years)
+6" 40 sin (0202t+231°2) (period 1780 years),
will produce inequalities with the same arguments due to indirect. action having the approximate coefficients
-004 and -0′′‘9
respectively. These results are sufficiently accurate for tabular purposes, but I shall give a more complete computation, with an examination of the whole effect of the terms of the second order.
New Haven, Conn. U.S.A.: 1907 November 26.
Postscript.-Since this paper was sent in, I have computed the terms due to the motion of the ecliptic and have discovered a few new inequalities containing the arguments of the planets. The most important is one with a coefficient o" 21 due to Jupiter, and having a period of 280 years.
1908 January 10.
Note on the Single Equation which comprises the Theory of the Fundamental Instruments of the Observatory. By Sir Robert Ball, LL.D., F.R.S.
We may conceive a generalised astronomical instrument of which the essential parts are as follows:
There is a fundamental axis, which we shall distinguish as axis I. It is capable of rotation in fixed bearings, and to it is attached an index which points to a reading R on a fixed graduated circle A. Axis I passes through the centre of A, and is normal to the plane of A.
Axis II is capable of rotation in bearings fixed on axis I. second graduated circle B is attached to axis II which passes through the centre of B, and is normal to its plane. The reading of B is R', as shown by an index rigidly attached to I. It may be observed that an index parallel to the intersections of the planes of A and B will serve for reading both circles, and the geometry of the question is simplified by employing this index.
It is necessary to distinguish between the two poles on the celestial sphere which are defined by the plane of a graduated circle. From one of these poles the graduation would appear to increase clockwise. From the other pole the graduation would appear to increase anti-clockwise. It is the latter pole which we shall here employ. The angle between I and II is the angle (180°) between the poles of A and B. We shall express it by
90° - y.
The telescope is rigidly attached to axis II, and when the optical axis of the telescope is directed to a star, the arc (→ 180°) from that star to the pole of B is also a constant of the instrument. We shall denote it by 90°+r.
The semiplane through axis II and that half of the telescope which contains the objective, cuts B at the graduation we shall
Let R1, R and R, R' be the readings of the instrument when directed successively to stars S, and S, with celestial co-ordinates a, and a 82. These co-ordinates may be altitude and azimuth, or right ascension and declination, or latitude and longitude, or any system in which the fundamental circles are rectangular. Then the equation we desire is obtained by equating two different