Neptune was discovered in 1846 as a result of mathematical calculation, done independently and practically simultaneously by Adams and le Verrier. The full story abounds in unexpected twists, and is complicated by personal matters, some of them rather painful. There is a fascinating account in Professor W. M. Smart's John Couch Adams and the Discovery of Neptune, published by the Royal Astronomical Society, 1947. I am concerned only with limited parts of the field.
To refresh the reader's memory of what has been said from time to time about the discovery I will begin with some representative quotations. In The Story of the Heavens (1886) Sir Robert Ball has passages:' the name of le Verrier rose to a pinnacle hardly surpassed in any age or country' ... 'profound meditations for many months' ... 'long tand arduous labour guided by consummate mathematical artifice'. The author is not above a bit of popular appeal in this book —— 'if the ellipse has not the perfect simplicity of the circle, it has at least the charm of variety ... an outline of perfect grace, and an association with ennobling conceptions' —— but on Neptune he is speaking as a professional. An excellent modern book on the history of astronomy has, so late as 1938:' probably the most daring mathematical enterprise of the century ... this amazing task, like which nothing had ever been attempted before'.
The immediate reaction was natural enough. Celestial Mechanics in general, and the theory of perturbations in particular, had developed into a very elaborate and high-brow subject; the problem of explaining the misbehaviour of Uranus by a new planet was one of 'inverse' theory, and the common feeling was that the problem was difficult up to or beyond the point of impossibility. One might speculate at some length on reasons for this opinion (one, perhaps was confusion between different meanings of the technica term 'insoluble'1). When Adams and le Verrier provec the opinion wrong (and after all any mathematical proof is a debunking of sorts) there was still something to be said for the principle that difficulties are what they seem before the event, not after. Certainly no one would grudge them their resounding fame. (Nor grudge, at a lower level, the luck of a discovery which makes a more sensational impact than its actual difficulty strictly merits; in point of fact this luck never does happen to the second-rate.) If the discovery has had a very long run one must remember that there is a time-lag ; people cannot be always reconsidering opinions, and having said something once even the most intelligent tend to go on repeating it. The phrase was still in vogue that 'only 3 people understand Relativity' at a time when Eddington was complaining that the trouble about Relativity as an examination subject in 'Part III' was that it was such a soft option.
In what I am going to say I am far from imputing stupidity to people certainly less stupid than myself. My little jeuz d'esprit are not going to hurt anyone, and I refuse to be deterred by the fear of being thought disrespectful to great men. I have not been alone in a lurking suspicion that a much simpler approach might succeed. On the one hand, aim at the minimum needed to make observational discovery practicable; specifically at the time \(t_{0}\) of conjunction.2 On the other, forget the high-brow and laborious perturbation theory, and try 'school mathematics'. (I admit to the human weakness of being spurred on by the mild piquancy success would have.) To begin with I found things oddly elusive (and incidentally committed some gross stupidities). In the end an absurdly simple line emerged: I can imagine its being called a dirty trick, nor would I deny that there is some truth in the accusation. The only way to make my case is to carry out the actual 'prediction' of \(t_{0}\) from the observational data, with all the cards on the table (so that anyone can check against unconscious or conscious faking). I will also write so as to take as many amateurs as possible with me on the little adventure.
A planetary 'orbit' is an ellipse with the Sun \(S\) at a focus, and the radius vector \(SP\) sweeps out area at a constant rate (Kepler's second law). An orbit, given its plane, is defined by 4 elements, \(a\), \(e\), \(\alpha\), \(\epsilon\). The first 3 define the geometrical ellipse: \(a\) is the semi-major axis; \(e\) the eccentricity; \(\alpha\) the longitude of perihelion, i.e. with the obvious polar coordinates \(r, \theta\), \(\theta\) is the 'longitude' and \(\theta=a\) when \(P\) is nearest \(S\) (at an end of the major axis). When we know \(a\) we know the 'mean angular velocity' \(n\) and the associated period \(p= 2 \pi/n\); \(n\) is in fact proportional to \(a^{-\frac{1}{2}}\) (Kepler's third law)3; further the constant rate of area sweeping is \(\frac{1}{2}abn\)4 and twice this rate is identical with the angular momentum'5 (a.m. for short); this has the differential calculus formula \(r^{2}\theta\), and it also is of course constant. The 4th element, the 'epoch' \(e\), is needed to identify the origin of \(t\); the exact definition is that \(\theta= \alpha\) (perihelion) occurs at the \(t\) for which \(nt+ \epsilon = \alpha\).
\(U\)'s orbit has a period of 84 years, and an eccentricity \(e\) of about \(\frac{1}{20}\). The effects of bodies other than \(S\) and \(N\) can be allowed for, after which we may suppose that \(U\), \(S\), and the eventual \(N\) are the only bodies in the system; we may also suppose (all this is common form) that the movements are all in one plane. The values of \(\theta\) (for \(U\)) at the various times \(t\) (we sometimes write \(\theta(t)\) to emphasize that \(\theta\) is 'at time \(t\)') may be regarded as the observational raw material (though of course the actual raw observations are made from the Earth). The \(r\)'s for the various \(t\) are indirect and are much less well determined.
The position in 1845 was that no exact elliptic orbit would fit the observed 6 over the whole stretch 1780 to 1840.6 The discrepancies are very small, mostly a few seconds of arc (with a sudden swoop of about 90",see Tablel). The ratio \(m\) of \(N\)'s mass to that of \(S\) (taken as 1) is actually about 1/19000 (the Orders of magnitude fit since m radians is about 11").
In the absence of \(N\) the a.m. \(A\) is constant (as stated above alias of Kepler's second law); the actual \(N\) accelerates \(A\) at times earlier than \(t_{0}\) and decelerates it at later times. The graph of \(A\) against \(t\) therefore rises to a maximum at \(t=t_{0}\), and my first idea was that this would identify. So it would if all observations were without error (and the method would have the theoretical advantage of being unaffected by the eccentricities). But the value of \(A\) at time \(t\) depends on the \(r\) at time \(t\), and the determinations of the \(A\)'s are consequently too uncertain. Though the method fails it rises from the ashes in another form. For this a few more preliminaries are needed.
The numerical data Adams and le Verrier had to work on were not the observed \(\theta\)'s themselves, but the differences between the observed \(\theta(t)\) and the \(\theta_{B}(t)\) of an elliptic orbit calculated by Bouvard; the 'discrepancy' \(\delta(t)\) (\(\delta\) for short) is \(\delta(t)=\theta(t)-\theta_{B}(t)\). [\(\theta_{B} (t)\) depends on the 'elements' of \(E_{B}\), and these are subject to 'errors'. These errors are among the unknowns that the perturbation theory has to determine: our method does not mind what they are, as we shall see.] Table I gives the raw \(S\)'s (given in Adams's paper7), together with the values got by running a smooth curve. The treatment of the start of the sudden swoop down after the long flat stretch is a bit uncertain: I drew my curve and stuck to it (but faking would make no ultimate difference). The differences show up the order of the observational errors (which naturally improve with the years something seems to have gone badly wrong in 1789); these are absolute, not relative (thus the probable absolute error in \(\delta_{1} - \delta_{2}\) is the same whether \(\delta_{1} - \delta_{2}\) is 0.5" or 90"). It is worth while to work to 0.1" and to the number of decimal places used in what follows, even though the last place is doubtful.
| Year | Observed \(\delta\) | Smooth Curve | Year | Observed \(\delta\) | Smooth Curve |
|---|---|---|---|---|---|
| 1780 | 3.5 | 3.5 | 1813 | 22.0 | 22.8 |
| 1783 | 8.5 | 8.5 | 1816 | 22.9 | 22.5 |
| 1786 | 12.4 | 12.5 | 1819 | 20.7 | 22.0 |
| 1789 | 19.0 | 15.8 | 1822 | 21.0 | 21.0 |
| 1792 | 18.7 | 18.3 | 1825 | 18.2 | 18.1 |
| 1795 | 21.4 | 20.3 | 1828 | 10.8 | 10.3 |
| 1798 | 21.0 | 21.6 | 1831 | -4.0 | -4.0 |
| 1801 | 22.2 | 22.4 | 1834 | -20.8 | -20.8 |
| 1804 | 24.2 | 22.8 | 1837 | -42.7 | -42.5 |
| 1807 | 22.1 | 23.0 | 1840 | -66.6 | -66.6 |
| 1810 | 23.2 | 23.0 | 1843(e) | \(-\) | -94.0 |
Table 1
The value for 1843 is an extrapolation; results derived from it are labelled '(e)'.
An 'effect' due to \(N\) is of 'order \(m\) in mathematical notation \(O(m)\); if, for a particular quantity \(X\), X$ denotes (calculated \(X\)) —— (observed \(X\)), then any \(\Delta X\) is \(O(m)\). The square of this (2nd order of infinitesimals) is extremely minute and everyone neglects it instinctively (if \(a\) watch loses 10 seconds a day you don't try to correct for the further loss over the lost 10 seconds the cases are comparable). Next, an effect of \(N\) is what it would be if \(U\), and also \(N\), moved in circles, plus a 'correction' for the actual eccentricities of the orbits. \(U\)'s eccentricity \(e(\frac{1}{20})\) is unusually large and it would be reasonable to expect \(N\)'s to be no larger (it is actually less than \(\frac{1}{100}\)). The \(e\)'s distort the 'circular' value of the effect by 5 per cent, (or say a maximum of 10 per cent.); the 'distortion' of the effect is \(O(em)\), the effect itself being \(O(m)\). I propose to ignore things of order \(O(em)\)8: this is the first step in my argument. In particular, when we have something which is either some \(\Delta\), or \(m\) itself, multiplied by a factor, we can substitute first approximations (i.e. with \(e=0\)), or make convenient changes that are \(O(e)\), in the factor.
Suppose now that \(E_{l}, E_{2}\) are two (exact) elliptic orbits, yielding \(\theta(t)\)'s that differ by amounts of the kind we are concerned with, differing, that is, by \(O(m)\).9 It is now the case that the differences satisfy the equation
\[\begin{equation} \theta_{1} - \theta_{2} = m( a + bt + c \cos nt + d \sin nt) + O(em) \end{equation}\]
where \(a, b, c, d\) are constants depending on the two sets of elements of \(E_{1}, E_{2}\), and (following our agreement about factors of \(m\)) \(n\) is any common approximation to the mean angular velocity. I will postpone the school mathentatics proof of this.
Next, (i) let \(E^{*}\) be the 'instantaneous orbit at time \(t_{0}\)', that is to say the orbit that \(U\) would describe if \(N\) were annihilated at time \(t_{0}\): note that \(E^{*}\) shares with \(t_{0}\) the property of being 'unknown'. (ii) Let \(\vartheta\) be the perturbation of the \(\theta\) of \(U\) produced by \(N\) since time \(t_{0}\).10 Then if, at any time \(t\), \(\theta\) is (as usual) \(U\)'s longitude, \(\theta_{B}\) is the longitude in the orbit \(E_{B}\), and \(\theta^{*}\) the longitude in the orbit \(E^{*}\), we have \(\vartheta = \theta - \theta^{*}\), and so
\[\begin{equation} \delta(t) = \theta - \theta_{B} = (\theta^{*} - \theta_{B}) + \vartheta \end{equation}\]
Now everything in this has a factor \(m\), and we may omit any stray \(O(em)\)'s. In particular, we may in calculating \(\vartheta\) drop any \(e\) terms. But this means that we can calculate \(\vartheta\) as if both \(U\)'s and \(N\)'s orbits were circles. When, however, the orbits are circles, \(\vartheta\) has equal and opposite values at \(t\)'s on equal and opposite sides of \(t_{0}\); in other words, if we write \(t=t_{0} + \tau\), then
\[\begin{equation} \vartheta(t) = \Omega (\tau) \end{equation}\]
where \(\Omega (\tau)\) is an odd11 function of \(\tau\); i.e. \(\Omega(-\tau)= -\Omega(\tau)\).
This, used in combination with (1) and (2), is the essential (and very simple) point of the argument. The difference \(\theta^{*} - \theta_{B}\) is a special case of \(\theta_{1} - \theta_{2}\) in (1). Write \(t=t_{0} + \tau\) in (1) and combine this with (2) and (3); this gives, ignoring \(O(em)\)'s,
\[\delta(t_{0} + \tau) = m \left\{a + b t_{0} + b \tau + c \cos (nt_{0} + n \tau) + d \sin (nt_{0} + n \tau) \right\} + \Omega(\tau)\]
Expanding the \(\cos\) and \(\sin\) of sums and rearranging we have (with new constants, whose values vary with but do not concern us)
\[\delta(t_{0} + \tau) = A - B(1 - \cos n\tau) + \left\{C \tau + D \sin n\tau + \Omega(\tau)\right\}\]
The curly bracket is an odd function of \(\tau\). Hence if we combine equal and opposite \(\tau\) and construct \(\delta^{*}(\tau)\) and \(\rho(\tau)\) to satisfy
\[\delta^{*}(\tau) = - \frac{1}{2} \left\{\delta(t_{0}+\tau) + \delta(t_{0}- \tau) - 2 \delta(t_{0}) \right\}, \rho(\tau) = \delta^{*}(\tau)/(1-\cos n\tau)\]
we have \(\delta^{*}(\tau)=B( 1 - \cos n\tau)\), and so $()=B $ for all \(\tau\). If, then, we are using the right \(t_{0}\) the ratio p(r) must come out constant: this is our method for identifying \(t_{0}\). The actual value of \(t_{0}\) to the nearest year is 1822.
Table II, in which the unit of time is 1 year (and the \(n\) of \(\cos n\tau\) is \(2\pi/84\)), shows the results of trying various (the century is omitted from the dates). The last place of decimals for the \(\rho(\tau)\) is not reliable, but of course gets better as the size of the entry \(2\delta^{*}(\tau)\) increases: I give the numbers as they came, and they speak for themselves. \(\tau=6\) is included, though the proportionate error in \(\delta^{*}\) is then considerable.12 For \(t_{0}=13\) \(\rho\) goes on to 34.8 at \(\tau=27\); for \(t_{0}=16\) it goes to 38.2 at \(\tau= 24\). Once the data the smooth curve values were assembled the calculations took a mere hour or so with a slide-rule. The date 1822-4 seems about the 'best' \(t_{0}\).
| \(t_{0} = 13\) | \(t_{0} = 16\) | \(t_{0} = 19\) | \(t_{0} = 22\) | \(t_{0} = 22.4\) | \(t_{0} = 25\) | \(t_{0} = 28\) | ||||||||
| \(\tau\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) | \(2\delta^{*}(\tau)\) | \(\rho(\tau)\) |
| 6 | 0.6 | 3.0 | 1.0 | 5.1 | 3.6 | 18.3 | 9.2 | 47.0 | 10.6 | 53.8 | 18.2 | 92.5 | 20.4 | 103 |
| 9 | 1.8 | 4.1 | 3.9 | 9.0 | 10.7 | 24.6 | 23.2 | 53.3 | 25.0 | 57.6 | 34.5 | 79.3 | 41.1 | 94 |
| 12 | 5.3 | 7.6 | 11.9 | 15.8 | 25.0 | 33.2 | 39.8 | 53.0 | 42.2 | 56.1 | 55.9 | 74.4 | 64.7 | 86 |
| 15 | 13.7 | 12.1 | 26.6 | 23.5 | 42.0 | 37.1 | 61.5 | 54.4 | 64.0 | 56.5 | 79.8 | 70.5 | 91(e) | 80.6(e) |
| 18 | 29.3 | 18.8 | 44.2 | 28.4 | 64.1 | 41.2 | 85.8 | 55.2 | 88.6 | 57.0 | 106(e) | 86.3(e) | \(-\) | \(-\) |
| 21 | 48.3 | 24.1 | 67.2 | 36.6 | 89.0 | 45.5 | 113(e) | 56.5(e) | 116(e) | 58.0(e) | \(-\) | \(-\) | \(-\) | \(-\) |
We need fairly large \(\tau\) for \(\delta^{*}(\tau)\) to have enough ignificant figures, and further to provide a range showing up whether \(\rho(\tau)\) is constant or not. And we need room to manoeuvre round the final. So the method depends on the 'luck' that 1822 falls comfortably inside the period of observation 1780-1840. But some luck was needed in any case.
It is an important point that the method is quite indifferent to how well \(E_{B}\) does its originally intended job, and we do not need to know (and I don't knoiv) its elements; it is enough to know the 'discrepancy' with some, 'unknown', orbit (not too bad of course). On the other hand the method ostentatiously says nothing at all about the mass or distance of \(N\). I will add something on this. With \(e\)-terms ignored \(\vartheta(\tau)/m\) can be calculated exactly for any given value of \(\lambda=a/a_{1}\) (ratio of the \(a\)'s of \(U\) and \(N\))13 The idea would be to try different \(A\)'s, each \(A\) to give a best fitting \(m\), and to take the best fitting pair \(\lambda, m\). This fails, because the greater part of \(\vartheta\) is of the form \(b^{\prime}(n\tau - \sin n\tau)\), and \(b^{\prime}\) is smothered by the \(a, b, c, d\) of \(\theta^{*} - \theta_{B}\), which depend on the unknown elements of \(E_{B}\) (\(\vartheta\) is smothered by the 'unknown' \(\theta^{*} - \theta_{B}\)). If we knew these elements (or equivalently the raw \(\theta\)) we might be able to go on. They could be recovered from the Paris Observatory archives; but this article is a last moment addition to the book, I do not feel that I am on full professional duty, and in any case we should be losing the light-hearted note of our adventure.
The time \(t\) once known, it would be necessary to guess a value for \(N\)'s distance \(a_{1}\); \(N\)'s period is then \(84(a_{1}/a)^{\frac{3}{2}}\) years, and we could 'predict' \(N\)'s place in 1846. The obvious first guess in 1846 was \(a_{l}/a=2\), following Bode's empirical law, to which N is maliciously the first exception, the true value being 1.58. Adams and le Verrier started with 2 (Adams coming down to 1.942 for a second round). Since from our standpoint14 too large an \(a_{1}\) has disproportionately bad results as against one too small, it would be reasonable to try 1.8. This would give a prediction (for 1846) about 10\(^{\circ}\) out, but the sweep needed would be wholly practicable.
Le Verrier was less than 1\(^{\circ}\) out (Adams between 2\(^{\circ}\) and 3\(^{\circ}\)); 'they pointed the telescope and saw the planet'. This very close, and double, prediction is a curiosity. All the observations from 1780 to 1840 were used, and on an equal footing, and the theory purported to say where \(N\) was over this whole stretch. With a wrong \(a_{1}\) they could be right at 1840 only by being wrong at 1780. With Adams's \(a_{1}=1.94a\) \(N\)'s period (which depends on \(a_{1}\) only) would be 227 years; he would have been wrong by 30\(^{\circ}\) for 1780 if the orbit were circular, and so the angular velocity uniform. But faced with a wrong \(a_{1}\) the method responded gallantly by putting up a large eccentricity (\(\frac{1}{8}\)), and assigning perihelion to the place of conjunction. The combination makes the effective distance from \(S\) over the critical stretch more like \(1.7 a_{l}\), and the resulting error at 1780 (the worst one) was only 18. (A mass 2.8 times too large was a more obvious adjustment.)
In much more recent times small discrepancies for \(N\) and \(U\) (\(U\)'s being in fact the more manageable ones) were analysed for a trans-Neptunian planet, and the planet Pluto was found in 1930 near the predicted place. This was a complete fluke: Pluto has a mass probably no more than \(\frac{1}{10}\) of the Earth's; any effects it could have on \(N\) and \(U\) would be hopelessly swamped by the observational errors.
It remains for me to give the (school mathematics) proof of (1) above. Call \(e_{1} - e_{2}\) \(\Delta e\), and so on. I said above that all \(\Delta\)'s were \(O(m)\): this is not quite true, though my deception has been in the reader's best interests, and will not have led him astray.15 It is true, and common sense, for \(\Delta a\), \(\Delta e\), \(\Delta n\), and \(\Delta \epsilon\). But the 'effect' of a given Aa vanishes when \(e=0\), and is proportional to \(e\). So it is \(e\Delta \alpha\), not \(\Delta \alpha\), that is comparable with the other \(\Delta\)'s and so \(O(m)\).16
We start from two well-known formulae. The first is geometrical; the polar equation of the ellipse of the orbit is
\[\begin{equation} r = a(1- e^{2}) ( 1 +e \cos (\theta -a))^{-1} \end{equation}\]
The second is dynamical; the equation of angular momentum (Kepler's second law) is
\[\begin{equation}r^{2} \frac{d\theta}{dt} = na^{2} (1-e^{2})^{\frac{1}{2}}\end{equation}\]
So, using dots for time differentiations,
\[\begin{equation} \theta = n(1-e^{2})^{-\frac{3}{2}} [1 - 2\cos (\theta -a) + 3e^{2} \cos^{2} (\theta -a ) + \cdots] \end{equation}\]
The first approximation (with \(e=0\)) is \(\theta=nt + \epsilon\). We take suffixes 1 and 2 in (6) and operate with \(\Delta\), remembering that we may take first approximations in any factor of an \(m\).
In estimating \(\Delta\theta\) we may, with error \(O(em)\), ignore the factor \((1-e^{2})^{-\frac{3}{2}}\) in (6), since it is itself \(\left(1+O(e^{2})\right)\), and its \(\Delta\) is \(O(e\Delta e)=O(em)\). We have, therefore, with error \(O(em)\),
\[\Delta \theta = \Delta \left\{n[\quad ]\right\} = [\quad ] \Delta n + n \Delta [\quad ]\]
The 1st term is \(\Delta n+O(em)\). The 2nd is
\[n\left[\Delta e \left\{- 2 \cos(\theta -a ) + O(e)\right\} + \Delta (\theta -a ) \left\{2e \sin (\theta -a ) + O(e^{2}\right\}\right]\]
and we may drop the \(\theta\) in \(\Delta (\theta - a)\) on account of the factor \(O(\epsilon)\). Summing up, we obtain
\[\Delta \theta = m (A+B \cos (\theta -a ) + C \sin (\theta -a ) ) + O(em)\]
where \(mA=\Delta n\), \(mB = -2n\Delta e\), \(mC = -2n(e\Delta a)\). Substituting the first approximation \(\theta=nt + \epsilon\) in the right hand side, we have
\[\Delta \theta = m (A+B \cos (nt + \epsilon -a ) + C \sin (nt + \epsilon -a ) ) + O(em)\]
and integration then gives
\[\Delta \theta = m (A+B/n \sin (nt + \epsilon -a ) - C/n \cos (nt + \epsilon -a ) ) + O(em)\]
which, after expanding the sin and cos and rearranging, is of the desired form (I).17
Its attachment to the '3-body problem' misleads people to-day.↩︎
The time at which NUS is a straight line (I shall use the abbreviations S, U, N).↩︎
It does not depend on \(e\).↩︎
The total area of the ellipse is \(\pi ab\), and it is swept out in time \(p\).↩︎
Strictly speaking the a.m. should have the planet's mass as a factor: but \(U\)'s mass is irrelevant and I omit it throughout.↩︎
Observations after 1840 were not immediately available, and anyhow were not used. Uranus was discovered in 1781. Lest the reader should be worried by small inconsistencies in my dates I mention that 1780 is 'used', the extrapolation being a safe one.↩︎
Collected Works, I, p. 11. These (and not the modifications he introduces, which are what appear in Smart) are what is relevant for us.↩︎
I should stress that there is no question of ignoring even high powers of \(e\) unaccompanied by a factor \(m\) (\(e^{4}\) radians is about 1"). The distortion in the value found for \(t\) is, however, a sort of exception to this. But the effect of \(e\)'s in distorting \(t_{0}\) is unlikely to be worse than the separation they create between time of conjunction and time of closest approach. An easy calculation shows that this last time difference is at worst 0-8 years.↩︎
The orbits may have 'Suns' of masses differing by \(O(m)\).↩︎
We allow, of course, negative values of \(t - t_{0}\) both in \(E^{*}\) and in \(\vartheta\).↩︎
'\(\Omega\)' is a deputy for '\(O\)' (initial of 'odd'), which is otherwise engaged. The italicized statement in the text is true 'by symmetry': alternatively, reverse the motions from time \(t_{0}\). (The argument covers also the 'perturbation of 'the Sun' which is not so completely negligible as might be supposed.)↩︎
And the, values for \(\tau=6\) at \(t_{0}=22, 22.4\) are more uncertain than usual because of a crisis in the smooth curve.↩︎
From two second order differential equations. The formula involves 'quadratures', but in numerical calculation integration is quicker than multiplication. It would be comparatively easy to make a double entry table for \(\vartheta(\tau, \lambda)/m\)↩︎
Perturbation theory calculations have necessarily to begin by guessing \(a_{1}\); our guess need only be at the end.↩︎
'Wen Gott betrugt is wohl betrogon.'↩︎
This twist makes the 'obvious' approach of using the weil known expansion \[\theta = nt +\epsilon + 2e \sin (nt + \epsilon -a) + \frac{5}{4}e^{2} \sin 2 (nt + \epsilon -a ) + \cdots\] slightly tricky; we should have to keep the term in \(e^{2}\). The line taken in the text side-steps this.↩︎
We have treated \(\Delta n\) and \(\Delta a\) as independent (the latter happens not to occur in the final formula for \(\Delta \theta\)): this amounts to allowing different masses to the two 'Suns'. The point is relevant to certain subtleties, into which I will not enter.↩︎