A. MATHEMATICS AND ASTRONOMY. 25 



one in a perfectly straight line. That they are intimately 

 connected, however, there can be no doubt. The question is 

 as to the nature of their orbits ; and, with respect to this, 

 Flammarion states emphatically that they do not revolve 

 about each other. Is such a conclusion contrary to the law 

 of universal gravity ? At first sight it would seem to be so ; 

 and we may have therefore in these stars an exceptional 

 case, instead of that powerful proof of universal gravity 

 which was formerly supposed. But as, theoretically, it seems 

 unwise to doubt the universality of this law, we are led to 

 conclude that the two components of 61 Cygni form, in fact, 

 possibly only a double companion to a more distant princi- 

 pal star, unless, indeed, one of them is at a great distance 

 directly behind the other. The latter conclusion is favored 

 by Flammarion, who cites several very similar cases of large 

 proper motions in relative right lines. 6 ^, LXXX., 171. 



THEORY OF PLANETARY PERTURBATIONS. 



In the Analyst for November, 1875, Mr. George W. Hill, 

 of Nyack, New York, published an interesting and impor- 

 tant mathematical paper on the development of the per- 

 turbative function in periodic series. He states that of 

 the two methods ordinarily employed for developing this 

 function, in one the numerical values of the elements in- 

 volved are employed from the outset, and the results ob- 

 tained belong only to the special case that is thus treated. 

 In the other method all the elements are left indeterminate, 

 and thus is obtained a literal development possessing as 

 much generality as possible. It is with the latter method 

 that his own investigations have to do, and his object has 

 been to simplify as far as possible the formulae that were 

 published by Puiseux in Lionville's Journal for 1860, and 

 those obtained by Bourget, as published by him in the same 

 journal for 1873. He seeks to effect this object by the in- 

 troduction of the Besselian functions. Mr. Hill's thorough 

 acquaintance with the analytical treatment of the problem 

 in hand gives to his little memoir of twenty pages a very 

 high theoretical value, and renders it the most important 

 mathematical paper that has appeared in the Analyst since 

 the beginning of its publication ; and the appearance of a 

 few more memoirs of this stamp should, we think, render the 



B 



