246 report — 1859. 



20. Gauss s Fourth Demonstration. — The fourth and sixth demonstrations, 

 though somewhat different from one another, are both intimately connected 

 with the theory of the division of the circle. They must, therefore, be re- 

 garded as less direct than the earlier proofs, but they have contributed even 

 more to the methods and resources of the higher arithmetic. 



The fourth depends on the formula 



l+r + r i + r 9 + +r&»-»*=a# te-»1 -vA7....(A) 



in which i represents (as throughout this Report) an imaginary square root 



/-. 2cr ^ 



of — 1 ; n is any uneven number, V n its positive square root, r=cos — + 



i sin — • Let the series 

 n 



l+r k +r ik +r ; " c + . . +r(»-D 2 * be denoted by \fr (k,n) ; 

 in the particular case in which n is a prime number, it is easy to see that 

 v^ (k, ri)=( — 1 $ (1, n). Further, p and q denoting two prime numbers, it is 

 found by actual multiplication of the two series \p (p, q) and \p (q, p) that 



+ to f) X * fe,)-* (1,«); that i, (£) (ft- jfrffiffft 



If we substitute for the functions \p their values given by the equation (A), 

 we find ( - ) ( ) =i > an equation which gives a rela- 



tion between f - ) and I ±- \ coincident with that assigned in Legendre's Law 



of Reciprocity. 



The equation (A) is not easy to demonstrate. It is not indeed difficult to 

 show that the sum of the series on the left-hand side is + V n when » = 1, 

 mod 4 ; and +i V n when n = 3, mod 4. But the determination of the am- 

 biguous sign in these values appears to have long occupied Gauss. He has 

 effected it in his memoir (the Summatio Serierum, &c.) by establishing the 

 equality 



l+ r -f-,.*4- r 9 ^ |-r("->) 2 =(r— r~ } )(r 3 — /— 3 )....(r»-2— r-»+ 2 ) (B), 



which he obtains by writing r for x, and n — I for m, in the series 



1 —xm (1 — x m ) (1 — x m ~ l ) _ (\ —x m ) (1 —x m ~ l ) (l—x m ~ 2 ) , 

 13* (l-x)(l-x 2 ) (l-x)(l-x 2 )(l-x 3 ) 



This series when m is a positive integer becomes an integral algebraical 

 function, and is proved by Gauss to be zero if m be uneven ; and if m be 

 even, to be equal to the product (l—x) (1 —x") ... (1 — x m ~ l ). From this 

 last observation, the demonstration of the formula (B) naturally flows. If n 

 be an uneven number, the formula (A) becomes 



l +r +r i +r°+ ... +H»-D 2 =(l+i)v / « or=0 (A') 



according as n is evenly or unevenly even. 



A very different, but a simpler demonstration of these formulae (A) and 

 (A'), depending on the properties of the definite integrals 



r+» /*+» _ /* +~ 



\ cos x* dx, I sinx^dx, or 1 e" 2 dx, 



J -M J -oo J -oo 





