their 
origin ; 
their 
amount. 
816 
no permanent alteration of the mean motion of 
either ; a conclusion which, as we have seen in the 
last section, he afterwards generalized for the pla- 
netary system. Several other memoirs by Lagrange 
and himself followed; and when the question be- 
came thus narrowed to periodic perturbations only, 
Laplace, with characteristic ardour and resolution, 
determined to search out every term which could affect 
the result; an irksome task, less congenial to the gene- 
ralizing spirit of Lagrange. He had already noticed, 
in his memoir of 1773, that Euler and Lagrange had, 
in their researches on this very subject, omitted terms, 
multiplied by sines and cosines of very small angles, 
which yet might, in the process of integration, be- 
come considerable by the largeness of the coefficients. 
Eleven years later he detected, in the expansion of 
the mutual perturbation of Jupiter and Saturn, terms 
of this kind.’ The coefficient (or maximum value of 
the term) is in this case divided by the square’ of the 
same quantity which renders the angle under the sine 
or cosine small. These terms were indeed likewise 
multiplied by the cubes of the excentricity, or like 
small quantities ; but notwithstanding this, by reason 
of the small divisor just mentioned, they were capable 
of attaining a formidable magnitude ; in the case of 
Jupiter, to 21’, and of Saturn, to 48’ or 49. That 
so small a force should produce so large an effect is 
due to the very long period of the most considerable 
portion of this inequality, which, in fact, led to its 
being confounded with perturbations properly secu- 
lar. The period of complete recurrence of the effects 
is about 920 years; and during half this time the 
motion of one planet is being constantly accelerated, 
and that of the other retarded ; during the other half 
the action is reversed. An effect continually in- 
creasing or diminishing for so long a time, and be- 
tween the two most massive of the planetary bodies, 
is evidently liable to become considerable. The 
maximum displacement of Jupiter and Saturn 
Laplace found by calculation to have occurred in 
1560, explaining the peculiarity above mentioned 
in the comparison of ancient with modern observa- 
tions. 
When we look to the physical cause of the large- 
ness of these particular perturbative terms, it is found 
to be this ;—that the period of revolution of Jupiter 
compared to that of Saturn, is almost as the num- 
bers 2 and 5 :—in other words to the near commen- 
surability of the mean motions, Were they exactly 
in proportion to these numbers, formidable and per- 
manent changes would possibly result in the orbits, 
As itis, the planets comeinto conjunction when Jupiter 
has completed 5 revolutions, and about ~;th more; 
Saturn 2 revolutions, and ~,th more. Consequently 
the point of conjunction travels round the circumfer- 
ence after about 44 conjunctions have occurred, which 
requires nearly 2700 years. But a little considera- 
MATHEMATICAL AND PHYSICAL SCIENCE. 
[Diss. VI. 
tion will show that conjunctions occur successively 
at three nearly equidistant points of the circumference ; 
consequently the two planets will have been presented 
to one another in every possible variety of configu- 
ration, when the point of conjunction has travelled 
‘one-third round the circumference, that is in about 
900 years. : 
The effect of this great improvement in the (68. 
Theory of Jupiter and Saturn was, that the most an- 
cient observations were completely reconciled with the 
modern, and the modern with one another; the errors 
of the tables were immediately reduced to one-tenth 
of their former amount, and soon after to much less. 
III. The third topic which I must shortlydiscuss in 69.) 
connection with the career of Laplace, is the Theory Theory of 
of the Tides. ae 
The Newtonian Theory of the Tides has been ex- 70.) 
plained in Mr Playfair’s Dissertation, but its progress Newton's. 
during the 18th century has not been advyerted to Tatts a 
It will be suf- the Equili- 
the continuation by Sir John Leslie. 
Ber- 
ficient to state here, that it was pursued into its conse- brium 
quences with ability and success by Daniel Bernouilli, T!°°"Y- 
who in 1740 shared a prize of the French Academy 
of Sciences on this subject, along with Euler, Mac- 
laurin, and Cavalleri, a Jesuit, the last a supporter of 
the Cartesian vortices. It was, perhaps, the conelud- 
ing honour paid to that once popular theory. 
The Tidal Theory of Newton and Bernouilli 
71.) 
presumes the earth to be at rest; and also the waters Its results, 
of the ocean to be at rest, and at every moment in a 
state of equilibrium between the force of gravity, 
tending to the earth’s centre, and the lesser forces 
tending towards the sun and moon. That a theory, 
founded on suppositions so far from the truth (not to 
mention the irregular distribution of sea and land on 
the earth’s surface), should in any manner or degree 
represent correctly what happens, may be matter of 
just surprise. The leading phenomena are however 
tolerably consistent with it; the dependence of the 
great tides on the moon’s position with respect to 
the meridian of the port; the spring and neap tides 
when the sun’s action and that of the moon conspire 
with or oppose one another; the priming and laggung 
of the tides depending on the displacement of thevertex 
of the compound ellipsoid due to the combined effect 
of the sun’s and moon’s attraction, depending therefore 
on the moon’s elongation from the sun ; the effects of 
the moon being in the nearer or remoter part of her 
orbit ; all these facts are indicated by the Equilibrium 
Theory (as it has been termed), and are also results 
of observation, The theory, however, does not give 
the true depth of tide, nor (except in casual instances) 
does the time of high and low water coincide with 
theory ; besides many minor imperfections. 
Laplace had the singular boldness to attempt 
the solution of a problem, which is more one of hydro- 
1 In consequence of a double integration in respect of the time. 
(72.) 
