144 REPORT— 1860. 



_L-^^D 1 X 1 (a) + D 2 X 2 (a)+... 4-D A _ 2 X A _ 2 («), mod X, 



where X* (a) represents the function S y-* k & s , and D/, denotes, as in 



s=0 



rf* log/(e p ) 

 art. 55, the differential coefficient — — • In this congruence the first 



coefficient alone is altered when /(a) is multiplied by a simple unit; and only 

 the even coefficients are altered when/(a) is multiplied by a real unit. Now 

 Dj is rendered congruous to zero by the condition/(a) =/(l), mod (1 — *) 2 ; 

 and M. Kummer has shown that, by multiplying /(a.) by a properly chosen 

 real unit, D 2 , D^ ...Da._3 may be similarly made to disappear, so that we 

 obtain 



-L^ = D 3 X 3 ( a ) + D 3 X s ( a )+...+Dx_ 2 Xx_ 2 («),modX, 



a congruence which is proved to involve the second congruence of condition 

 satisfied by a primary number, i. e.f(a)f(a.- i )=f(\y,mo& \*. 



58. The methods to which M. Kummer at first had recourse in order to 

 obtain a demonstration of his theorem, consisted in extensions of the theory 

 of the division of the circle. By such extensions he demonstrated the com- 

 plementary theorems, and even a particular case of the law of reciprocity 

 itself — that in which the two complex primes compared are conjugate. But, 

 after repeated efforts, he found himself compelled to abandon these methods, 

 and to seek elsewhere for more fertile principles. " I turned my attention," 

 he says, " to Gauss's second demonstration of the law of quadratic recipro- 

 city, which depends on the theory of quadratic forms. Though the method 

 of this demonstration had never been extended to any other than quadratic 

 residues, yet its principles appeared to me to be characterized by such 

 generality as led me to hope that they might be successfully applied to 

 residues of higher powers ; and in this expectation I was not disappointed t." 



M. Rummer's demonstration of the law of reciprocity was communicated 

 to the Academy of Berlin in the year 1858, ten years after the date of his 

 first discovery of it. An outline of the demonstration is contained in the 

 Monatsberichte for that year; and it is exhibited with great clearness and 

 fulness of detail in a memoir published in the Berlin Transactions for 

 1859, which contains what is for the present the latest result of science on 

 a problem which, if we date from the first enunciation of the quadratic 

 theorem by Euler, has been studied by so many eminent geometers for 

 nearly a century. It would, however, be impossible, without exceeding the 

 limits within which this Report is confined, to give an account of its contents, 

 which should be intelligible to persons not already familiar with the subject 

 to which it refers. Taken by itself the demonstration of the theorem is, indeed, 

 sufficiently simple ; but it is based on a long series of preliminary researches 

 relating to the complex numbers that can be formed with the roots of the 

 equation w K =D (z), in which D (a) itself denotes a complex number com- 

 posed of Xth roots of unity. To those researches, and to the demonstration 

 of the law of reciprocity founded on them, we shall again very briefly refer, 

 when we come to speak of the corresponding investigations in the theory of 

 quadratic forms, an acquaintance with which is essential to a comprehension 

 of the method adopted by M. Kummer in his memoir. We may add that 

 M. Kummer has intimated that he has already obtained two other demon- 



* Crelle, vol. xliv. p. 130-140. f See the Berlin Transactions for 1859, p 29. 



