18 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



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, thatEuler 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, 

 their Eleven years later he detected, in the expansion of 

 ori g in 5 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 1 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 

 their small quantities ; but notwithstanding this, by reason 

 amount. o f ^he sma ll 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. 



(67.) When we look to the physical cause of the large- 

 ?rne .ir 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 it is, the planets come into conjunction when Jupiter 

 has completed 5 revolutions, and about -faih 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- 



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 shortly discuss in (69.) 

 connection with the career of Laplace, is the Theory Theory of 

 of the Tides. 



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 adverted to in n o u jiij" 8 OI 

 the continuation by Sir John Leslie. It will be suf- the Equili- 

 ficient to state here, that it was pursued into its conse- brium 

 quences with ability and success by Daniel Bernouilli, Tueor y- 

 who in 1 740 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 conclud- 

 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 lagging 

 of the tides depending on the displacement of the vertex 

 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- 



In cons equence of a double integration in respect of the time. 



