340 



SCIENCE. 



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



and variety of these particular solutions is 

 far greater than one would at first sight 

 imagine. 



We come now to a second class of partic- 

 ular solutions named by the author asymp- 

 totic. It arises from the consideration of 

 solutions diifering very little from periodic 

 solutions. Here we have to deal with 

 linear differential equations having periodic 

 coefficients. The integrals of these contain 

 in their terms exponential factors, and on 

 the nature of the exponents of these factors 

 depends the quality of the resulting solu- 

 tions. M. Poincare has named these expo- 

 nents characteristic. They are roots of an 

 algebraic equation of a degree equal to the 

 number of dependent variables involved in 

 the question. If any of these roots are 

 imaginary with real portions or wholly real, 

 we are in presence of asymptotic solutions. 

 The algebraic equation mentioned contains 

 the unknown only in even powers ; hence 

 the characteristic exponents are in pairs 

 having the same absolute value, but with 

 contrary signs. In all the cases presented 

 by astronomy, where, on account of the 

 near approach to circular motion, a periodic 

 solution can be taken as a first approxima- 

 tion, it appears that the squares of the 

 characteristic exponents are all real and 

 negative. Thus, there is no call here to 

 consider this sort of solution, and this fact 

 must much diminish the interest of the 

 astronomer in it. M. Poincare has, how- 

 ever, elaborated it with great pains, show- 

 ing how the effect of higher powers of the 

 deviations from the periodic solution may 

 be taken into account. The series resulting 

 are, nevertheless, divergent, as in other 

 cases. 



The second volume of the Methodes Nou- 

 velles is devoted to the elaboration and con- 

 sideration of various processes for develop- 

 ing the integrals of planetary motion accord- 

 ing to the powers of a small parameter. The 

 chief of these are due to Professor Newcomb 



and MM. Lindstedt and Bohlin ; but M. 

 Poincare has augmented thenumber of them 

 by introducing modifications of his own. 

 All involve the principle of recurrence ; that 

 is, the first step is the only one which is in- 

 dependent, the following depend on all that 

 precede. These methods, in their general as- 

 pect, do not differ from the old developments 

 in powers of the disturbing force, except 

 the operations are so adjusted that the time 

 never escapes from the trigonometric func- 

 tions. This is accomplished by greatly aug- 

 menting the number of the elementary 

 arguments, and by supposing that the rate 

 of motion of each of these is developable 

 according to integral powers of the before- 

 mentioned parameter, or, in some cases, of 

 its square root. 



When there is more than one elementary 

 argument, the series obtained in all these 

 ways are pronounced to be generally diver- 

 gent in the rigorous sense of the word. M. 

 Poincare brings forward several methods of 

 proof of this. The first depends on the 

 presence of small divisors in the expressions 

 of the coefficients. However, when we do 

 not insist on developments in powers of a 

 parameter, this method of proof has no ap- 

 plication. Another method is derived from 

 the principle that two characteristic expo- 

 nents vanish for every uniform integral that 

 exists. But the integrals which necessitate 

 this conclusion must not only be uniform, 

 they must be valid for every possible case 

 of the problem. Now the integrals known 

 as those of the conservation of living forces 

 and of areas are of this nature ; but the in- 

 tegrals derivable from the series of De- 

 launay, Newcomb and Lindstedt are valid 

 only for a limited range in the values of the 

 linear variables. For instance, in the prob- 

 lem of the three bodies, if the deformation 

 of the triangle formed by these bodies is 

 such that we cannot find any two sides, one 

 of which sustains to the other an invariable 

 relation of greater to less, we cannot apply 



