ON THE THEORY Of NUMBERS. 263 



to the product of ¥(x) by at! exponential function, if /u be an imaginary 00m* 

 pie* number. (Compare the formulae of sect. 61 of the 'FundamentaNova.') 

 The difficulty occasioned by this imperfect periodicity of ¥(x) Eisenstein has 

 overcome by the introduction of the number t, which is supposed to repre- 

 sent a real even indeterminate integer. The formulae on which his proof 

 depends, are 



(i) y(»+*)«««*F(*) f 



(ii) F(ix) =/e"' i F(.r), 



OH) S££) =c ,-, e -n4 + ^). 



The symbol to which depends on x, but is independent of t, is different in 

 each of these equations : in the first, k is any complex integer ; in the third, 

 c is a numerical constant independent of a? and/^; p x a primary number 

 prime to t ; p its norm; and r the general term of thejo— 1 residues of p v 

 the sign of multiplication II extending to every value of r. These equa- 

 tions, the first two of which depend on the most elementary properties of the 

 function ¥(x) or H (see ' Fundamenta Nova,' loc. cit.), while the third is of a 

 more abstruse character, Eisenstein has established by methods which are 

 peculiar to himself, and which it would take us too far from our present sub- 

 ject to describe. They serve to replace the formulae 



f(v) = (j>(v + 2ku)); 0(w) = fy(t>); 



(l>(v) n(i-oV) 



in Eisenstein's earlier demonstration ; and lead to the conclusion 



p-l q-\ 



(SMSM 



c wt- 



10 still denoting some quantity independent of I. And since in tVs formula 

 t may have any even value prime to p 1 and g v it is impossible that e** should 

 have any value but that of one of the fourth roots of unity, so that we 

 have e wt2 =l ; which gives the law of Reciprocity. 



36. An algorithm has been given by Eisenstein* for calculating the value 



of the symbol ( | v^, ) by means of the development of - ^ ■ .^ in a con- 

 tinued fraction. This algorithm, in a slightly simplified form, is as follows: — 

 Let a + ia'^=p„, b + ib'^p^ and form the series of equations 



AMJWi+«* +, « 



The numbers p and p x are supposed to be uneven, and prime to one an- 

 other;/^ is primary; the quotients / .„, k v k 2 .. k t are all divisible by l+i', and 



* Crelle's Journal, vol. ixviii. p. 243. But the first inventiou of this algorithm, and of 



the similar one which exists in the Theory of Cubic Residues, is due to Jacob! (See the 

 note, " Ueber die Kreistheilung," &c, so often cited in this Report.) 



