154 Royal Astronomical Society. 



tween the author's results and those of Laplace is investigated with 

 the same clearness which prevails through the other operations. 



" In one respect, the plan of investigation differs much from 

 those of his predecessors, as well as from Hansen's. The investi- 

 gation is wholly symbolical : no numerical value is introfiuced, and 

 no consideration of relation of values entertained, till the final sub- 

 stitutions are made. As an example of theory, there can be no 

 doubt of the beauty of this process. As a subject for practical ac- 

 curacy, it may not be so certain whether it is advisable. The con- 

 vergence of the series is sometimes extremel}^ slow. As far as I can 

 observe, the accuracy of this method is exactly and properly that of 

 successive substitution : but, in various parts of the lunar theory 

 (in all places where the terms rise two orders by integration), the 

 method of successive substitution is not sufficient ; in fact, it is ne- 

 cessary to assume a term in order to find its correct value. Adopting 

 this method, however, the author has pushed it as far as, probably, 

 it will ever be carried. The whole is worked to the fifth order, and 

 some parts to the seventh order. 



" Finally, the author has determined from observations the prin- 

 cipal constants which require to be substituted in the symbolical ex- 

 pressions, and has substituted them, and has thus produced a set of 

 numerical expressions which may immediately be used for the for- 

 mation of lunar tables. 



" In terminating the remarks on the works of these two authors. 

 Plana and Hansen, I must again express my very great admiration 

 for both. But their merits are of very different kinds. The theory 

 of Hansen is undoubtedly of the higher order, but it can hardly yet 

 be said to be practical (at least in the lunar theory) : many years 

 will yet elapse before it will influence the lunar tables. The theory 

 of Plana is very good, and probably adequate in all respects : it is 

 eminently practical in form : it has already influenced the investi- 

 gations of other writers, and will probably soon influence the 

 tables." 



There is but one thing more to add to this clear and powerful 

 summary, and I will supply it by a quotation from the work itself : — 



" Je n'ai pu me faire aider par personne ; j'ai du traverser seul 

 cette longue chaine des calculs, et il n'est pas etonnant si par 

 inadvertence j'ai omis quelques termes qu'il fallait introduire pour 

 me conformer a la rigueur de mes propres principes." When we 

 look at the work itself there seems something almost awful in this 

 announcement. 



A very important memoir of M. Plana, on the theory of the pla- 

 netarj' perturbations, has adorned the Transactions of this Society. 

 The points of which it treats are miscellaneous, and some of them, 

 perhaps, not of the highest importance, except in one point of view, 

 and that, perhaps, the most important of all. Every one who is at 

 all conversant with these researches must be impressed with the 

 enormous interval which separates — I will not say the mere differ- 

 ential equations of the planetary motions — but their integrals after 

 much and intricate development — from the final numerical results 



