336 



SCIENCE. 



[N. S. Vol. III. No. 62. 



to Cauchy, and is a most remarkable con- 

 tribution to the science of mathematics. 

 Supposing that the parameters begin from 

 zero values, this criterion amounts to say- 

 ing that the moment the function, which 

 the series is to represent ceases to be holo- 

 morphic, or becomes infinite, that moment 

 the series ceases to be convergent. Conse- 

 quently, if a space, having as many dimen- 

 sions as there are parameters in the cascj be 

 conceived, and a surface be constructed in 

 it formed by the consensus of all the points 

 where the considered function ceases to be 

 holomorphic, then, provided the values of 

 the parameters define a point within this 

 surface, that is, on the same side where 

 lies the origin, the series will be convergent. 

 Generally this surface will be closed, and, 

 within it, the function will not take infinity 

 as its value. 



Without any mathematical reasoning the 

 propriety of the principle just enunciated 

 may be perceived. Since it is possible for 

 the series in powers and products to give 

 only one value for the function, the moment 

 the latter may have any one of several 

 values, the series fails to give them all ; and, 

 as there is no reason why any particular 

 value should be selected, the conclusion 

 must be that it does not represent any of 

 them. Also, it is easy to see that, when 

 the function takes infinity as its value, the 

 series fails to represent it. 



In applying this principle to the series in- 

 volved in the treatment of the problem of 

 many bodies by Delaunay's method, it ap- 

 pears, at first sight, as if we must have some 

 finite representation of the coefficients in 

 question in order to discover the particular 

 points at which they cease to be holomor- 

 phic, such, for instance, as is given bj' an 

 algebi'aic or transcendental equation. But 

 this is not imperative, as it is often possible 

 to make this discovery from certain recog- 

 nized properties of the function considered, 

 without being in possession of its form ex- 



plicitlj' or implicitly. It appears probable 

 that, in the class of cases considered, the 

 mentioned coeflicients can be represented 

 by multiple definite integrals, all taken be- 

 tween the limits and -, the independent 

 variables being those which have been 

 denominated angular. Such functions are 

 always holomorphic, provided that the ex- 

 pressions under the signs of integration are 

 themselves holomorphic between the men- 

 tioned limits. If the statement just made 

 be admitted, although it may be impossible 

 to write explicitly the mentioned expres- 

 sions, we may, nevertheless, be certain that 

 they remain holomorphic, provided that the 

 linear variables, which may be the same 

 as the para.meters considered, are so re- 

 stricted in their range of values that no 

 matter what values the angular variables 

 receive, no distance between any two 

 bodies of the system can vanish. Or, in 

 other words, that the R of Delaunay must 

 never become infinite. Thus it seems 

 probable that the conditions of convergence 

 for Delaunay's series are precisely identical 

 with those for the stabilitj' of motion of the 

 system. 



The series arising in Delaunay's method, 

 as applied to the moon, contain five para- 

 meters ; the number would be six were the 

 moon's mass not neglected. We should 

 also have six in the application of the method 

 to two planets moving about the sun ; how- 

 ever, should we employ the well-known 

 function 6,"' of Laplace, the number would 

 be reduced to five. It ought to be possible, 

 therefore, after the performance of a limited 

 number of operations, to assign limiting 

 values to these parameters, below which 

 the series would certainly be convergent. 

 This also involves the possibility of finding 

 limits to the errors committed by truncating 

 the series at a certain order of terms. 

 Again, provided the time is limited to a 

 certain interval, the capacity of these trun- 

 cated series for representing the coordinates 



