January 23, 1914] 



SCIENCE 



121 



l=a cos y + i cos 31/ + c COS 5j/ + • • • • 

 For this purpose he differentiates an even 

 number of times, obtaining thus the system 

 of equations 



= a cos y + 63" cos 3y + c.5« cos 5y 



+ ... (B = 2, 4, ...) 



Combining this wdth the preceding equa- 

 tion and putting 2/^0, he obtains an infi- 

 nite number of equations of first degree 

 with an infinite number of unknowns, a, 

 b, c, ■■ ■ . To solve these he uses the first 

 m equations to determine the first m un- 

 knowns, suppressing all the other un- 

 knowns, and finally determines their limit- 

 ing values as m increases indefinitely. 

 There is no time to point out the lack of 

 rigor. Fourier uses his mathematics with 

 the delightful freedom and naivete of the 

 physicist or astronomer who trusts in a 

 mathematical providence. 



This suggestive line of attack was not 

 followed up, and indeed could not be, prior 

 to the development of a theory of infinite 

 determinants. When such a system of 

 linear equations with an infinite number 

 of unknowns came again to the foreground, 

 the inciting cause was again a trigono- 

 metric series. I refer, as you know, to the 

 work of our own astronomer, Hill. In his 

 memoir on the ' ' Motion of the Lunar Peri- 

 gee" he had before him a differential equa- 

 tion of the following form, with numerical 

 coefficients : 



= 2.(, 



-|- Si cos 2t -1- 02 cos 4t + 



Assuming a solution in the form 



n=+ao 



w = e»«T 2 6„e'""' 



(which except for the factor e*" is only a 

 trigonometric series under another guise). 

 Hill obtains for the determination of c and 

 the in an infinite system of equations lin- 

 ear in the in. The elimination of the b„ 

 then gives c as the root of a certain infinite 



determinant, and then the values of the h„ 

 are also found by use of infinite determi- 

 nants. 



The importance of Hill's results at once 

 attracted the genius of Poincare whose at- 

 tention had, in fact, been previously drawn 

 by Appell to an infinite system of linear 

 equations. Poincare now proceeded to con- 

 sider the question of the convergence of in- 

 finite determinants, and in so doing laid a 

 sound foundation for a new mathematical 

 subject. In this new theory of infinite de- 

 terminants the central thought is the pas- 

 sage, under restrictions to be properly 

 ascertained, from a finite to an infinite sys- 

 tem of linear equations. This principle here 

 employed has been since applied in an even 

 more striking manner by Fredholm, who 

 was led through its use to his historic 

 solution of a class of integral equations. 

 In the theory of these equations the infinite 

 determinant plays an indispensable role. 

 A sixth influence of Fourier's series is thus 

 seen in the origin of a theory of infinite 

 determinants, also indirectly in the theory 

 of integral equations for which it has sup- 

 plied an important tool. 



The seventh and the last infiuence on 

 which I shall specifically dwell is more sub- 

 tile, not so easily pointed out or demon- 

 strated as some of the foregoing, but never- 

 theless one of the most far-reaching and 

 probdbly the most pervasive of all. The 

 physicist, astronomer, or mathematician has 

 again and again to expand an arbitrary or 

 assigned fimction into a series of functions, 

 the nature of which varies with the prob- 

 lem before one. When once the idea and 

 method of expressing an arbitrary function 

 in series of sines and cosines have been won, 

 they can be extended to other series of 

 functions, as for instance series of Bessel's 

 functions, zonal harmonics. Lame polyno- 

 mials, spherical harmonics. For such de- 

 velopments the trigonometric series with its 



