Maech 6, 1896.] 



SCIENCE. 



341 



the mentioned series. And here it is well 

 to note that the defect of convergence does 

 not arise from the application of the pro- 

 cesses of integration, but already exists in 

 the development of the perturbative func- 

 tion before integration commences. Thus 

 Delaunay's development of this function at 

 the beginning of his lunar theory is diver- 

 gent and illusory, unless we have the lunar 

 radius in apogee always less than the solar 

 radius in perigee, and that without regard 

 to the mode of expressing the coefficients. 

 Some of the particular integrals relied upon 

 by M. Poincare to establish the vanishing 

 of all the characteristic exponents, in case 

 we accept M. Lindstedt's series as valid, lie, 

 so to speak, on the boundary of the domain 

 in which these series are convergent. 



In the third place an appeal is made to 

 the alleged non-existence of analytic and 

 uniform integrals beyond those already 

 known. Were this non-existence clearly 

 established it would decide the question on 

 the side where M. Poincare has placed him- 

 self. But, at least as far as the non-exist- 

 ence of integrals of this nature in a limited 

 domain for the linear variables is concerned, 

 the proof given for it is quite defective. 

 This proof consists in ascertaining how 

 these integrals, supposing them to exist, 

 would behave should we attempt to derive 

 periodic solutions from them. It is difficult 

 to present this matter without the assist- 

 ance of algebraic formulas; nevertheless, it 

 may be attempted. Let there be a number 

 of equations whose left members are formed 

 by the product of two factors. When we 

 pass to a periodic solution, one of these fac- 

 tors becomes zero. What conclusion can we 

 ■ draw from each of the thus modified equa- 

 tions? Evidently one of two things : either 

 the remaining factor of the left member is in- 

 finite and the right member indeterminate, 

 or it is finite and the right member a vanish- 

 ing quantity . No w in case we are obliged to 

 accept the first conclusion, were it only but 



once, M. Poincare has demonstrated the 

 non-existence of integrals ; but, granting 

 that it is proper in every case to accept the 

 latter conclusion, the demonstration fails. 

 Now he declines to consider the latter 

 alternative, saying that he does not believe 

 that any problem of dynamics, presenting 

 itself naturally, occurs where the right 

 members of the mentioned equations would 

 all vanish. But it should be borne in mind 

 that, while they do not vanish in the 

 general equations, the adjustment of the 

 values of the linear parameters required by 

 the passage to a periodic solution may bring 

 about their vanishing. Thus, in the lunar 

 theory, a periodic solution is brought about 

 by making e^O, e=0, and ^'=0, the result is 

 the vanishing of every coeificient having 

 any of these quantities as a factor. 



M. Poincare appeals in another place to 

 the fact that the Linstedt series, if conver- 

 gent, would establish the non-existence of 

 asymptotic solutions. But this observation 

 is irrelevant for the reason that the do- 

 mains of the two things are quite distinct. 

 In any case where Lindstedt's series are 

 applicable there are no asymptotic solu- 

 tions, and where there are asymptotic solu- 

 tions Lindstedt's series would be illusory. 



We owe much to M. Poincare for having 

 commenced the attack on this class of ques- 

 tions. But the mist which overhung them 

 is not altogether dispelled; there is room 

 for further investigation. 



G. W. Hill. 



ADMISSION OF AMERICAN STUDENTS TO THE 

 FRENCH UNIVERSITIES. 

 The Conseil Superieur de '1' Instruction 

 Publique has issued a decree removing the 

 restrictions upon the admission of Ameri- 

 can and other foreign students to the French 

 universities and giving them a status sub- 

 stantially similar to that accorded by the 

 German universities. This important con- 

 cession by the French authorities is the 



