464 



NA TURE 



{Sept. 27, 1877 



the element under consideration may be found by direct 

 integration, supposing the elements to be constant in the 

 terms to be miegrated, and the mean longitudes only to 

 vary. Also the secular variation of the element con- 

 sidered, that is, the rate of variation of the element when 

 cleared of peiiodic inequalities, will be given by the secular 

 terms taktn alone. It the disturbing masses, however, 

 are not very small, this process is not sufticiently accurate, 

 and the peiiodic inequalities thus found can only be re- 

 garded as a first appro.ximation to the true values. In 

 order to find more correct values, we must substitute for 

 the elements in the second member of the equation their 

 secular parts augmented by the approximate periodic 

 inequaliiies before found. 



Now, if in any periodic term we increase any element 

 by a periodic inequality depending on a different argu- 

 men^, t^at is, involving different multiples of the mean 

 longitudes, the result will evidently be to introduce new 

 periodic terms which will involve the square of one of the 

 masses or the product of two of them as a factor. Simi- 

 larly, if in any periodic term any element be increased by 

 a periodic inequality depending on the same argument, 

 the result will also introduce new terms of the second 

 order which do not involve the mean longitudes, and 

 which therefore constitute new secular terms. These will 

 be particularly important if the inequality in question be 

 one of long period. Also in the secular terms the result 

 of increasing any element by a periodic inequality will be 

 to introduce a new periodic term depending on the same 

 argument. Lastly, it should be remarked that in finding 

 the periodic inequalities of any element by integration of 

 the corresponding differential equation, we must take into 

 account the secular variations of the elements which were 

 neglected in the first approximation. The new terms thus 

 introduced, like the others which we have just described, 

 will evidently be of the second order with respect to the 

 masses. 



If the disturbing masses be large, as in the case of the 

 mutual disturbances of Jupiter and Saturn, it may be 

 necessary to proceed to a further approximation, and thus 

 to obtain new terms, both periodic and secular, which 

 involve the cubes and products of three dimensions of the 

 masses. The number of combinations of terms which 

 give rise to these teims of the second and third orders is 

 practically unlimited, and the art of the calculator consists 

 in selecting those combinations only which lead to sen- 

 sible results. This is the chief cause of the great com- 

 plexity of the theories of the larger planets, and more 

 especially of those of Jupiter and Saturn. 



M. Leverrier lays it down as the indispensable condi- 

 tion of all progress that we should be able to compare the 

 whole of the observations of a planet with one and the 

 same theory, however great may be the length of time 

 over which the observations extend. In order to satisfy 

 this condition, he develops the whole of his formula 

 algebraically, leaving in a general symbolical form all the 

 elements which vary with the time, such as the excen- 

 tricities, the inclinations, and the longitudes of the peri- 

 helia and nodes. He treats in the same way the masses 

 which are not yet sufficiently known. 



All the work is given in full detail, and is divided as 

 far as possible into parts independent of each other, so 

 that any part may be readily verified. All the terms 

 which are taken into account are clearly defined, so that 

 if it should ever be necessary to carry on the approxima- 

 tions still further, it will be easy to do so without having 

 to brgm the investigation afresh. The whole work is 

 presented with such clearness and method as to make it 

 an admirab e model lor all similar researches. 



After the development of the disturbing functions, and 

 the formation of -the differential equations on which the 

 variations of the elements depend, the first step to be 

 taken is to determme by integration of these equations 

 She periodic inequalities of the elements of the orbits of 



Jupiter and Saturn which are of the first order with respect 

 to the masses. As we have already said, the expressions 

 of these periodic variations of the elements are given with 

 such generality that, in order to obtain their numerical 

 values at any epoch whatever, it is sufficient to substitute 

 the secular values of the elements at that epoch. The 

 calculation of the various terms under this general form is 

 very laborious, and it requires great and sustained atten- 

 tion in order to avoid any error or omission of importance. 

 On the other hand, by substituting from the beginning 

 the numerical values of the elements at a given epoch, 

 the calculation is rendered much shorter and admits much 

 more readily of verification ; but the result thus obtamed 

 only holds good for the given epoch, and is thus entirely 

 wanting in generality. 



In the determination of the long inequalities of Jupiter 

 and Saturn, the approximation is earned to terms which 

 are of the seventh degree with respect to the excentricities 

 and the mutual inclination of the orbits. In the next 

 place the terms of the first order in the secular variations 

 of the elements of the orbits are determined. After this 

 the periodic inequalities of the second order with respect 

 to the masses aie considered. These are determined in 

 the same form as the terms of the first order, in order that 

 their expressions may hold good for any epouh whatever. 

 The formulae relating to these terms are necessarily very 

 complicated. The coefficient belonging to a given argu- 

 ment depends, in general, on a great number of terms 

 which are classed methodically. Next are determined 

 the terms of the second order in the secular variations of 

 the elements of the orbits. Afterwards, M. Leverrier 

 takes into account the influence of the secular inequali- 

 ties on the values of the integrals on which the periodic 

 inequalities depend. The last part of this chapter is 

 devoted to the completion of the differential expressions 

 of the secular inequalities by the determination of certain 

 secular terms in the rates ol variation of the excentricities 

 and the longitudes of the perihelia, which are of ttie third 

 and fourth orders with respect to the masses. 

 {To be CO mi lined.) 



NOTES 



We record with sincere regret the death of Prof. Alphonse 

 Oppenheim.at Hastings, on the I7tliin3t. ; he died by his own hand 

 through grief at the death of his wife. Prof. Oppenheim is well 

 known for his numerous researches in organic chemistry. Foi merly 

 one of the professors of chemistry at tte University of Berlin, 

 he only a few months ago, as we recorded at the lime, had 

 accepted the chair of chemistry at the University of Miinster, in 

 Westphalia. Prof. Oppenheim was a frequent contributor to 

 this journal, and was much esteemed by a large circle of friends 

 in England. 



The death is announced, 'on the 17th insf., at the age of 

 seventy-seven, of Mr. W. IL Fox Talbot, F.R.S., the inventor 

 of the photographic process known as Talbotype, a name latterly 

 merged in the general name photography. Mr. Talbat was a 

 man of varied attainments and manifold work. He was educated 

 at Harrow and Cambridge, where he distinguished himself as a 

 Greek scholar: He took a delight in chemistry, and it was in 

 1833 that he seems to have conceived the idea of inventing some 

 process by which the beautiful pictures exhibited in a camera 

 lucida could be impressed and rendered permanent. He and 

 Daguerre seem to have brought their several processes to a satis- 

 factory result almost simultaneously, though Daguerre was the 

 first to announce his process, in 1839. Mr. Talbot lost no time 

 in communicating to the Royal Society the details of his own 

 process, though it was not till 1840 that he made the discovery 

 which " laid the foundation of the photographic art in its present 

 form." In 1842 Mr. Talbot was presented with the gold medal 

 of the Royal Society. He did not patent his discovery, but on 



