350 report — 1865. 



the sign of multiplication II extending to every system of values of y, y, and Tc. 

 Let 6=1 + -, tr representing a real positive quantity; we obtain 



(Table A, vi. art. 125). Again, if 2 is the greatest common divisor of y + 2k 

 and y ', and if a, b, 2' are determined by the equations 



y + 2k . y ' s , n 



while c and d are two numbers, of which d is uneven, satisfying the equa- 

 tion ad— bc=l, we find 



y(l+^j+2Jc 



c + d M + tty 



a+b 

 whence (Table A, iii.) 



r(l+i\+2fc 



if 21 + 1 ==dy, mod 2'. If we give to y, y, and 1* in succession all the $(n) 



ac- 



systems of values of which they are susceptible, W*V. crj ~ ) will 



. y ' 

 quire in succession $(«) values, which (except for particular values of <r) 



are all different ; tp~ s ( ^ ) will therefore also acquire the same 



number of different values ; i. e. 2 will represent in succession every divisor 

 of n, and2Z + l every residue of its conjugate divisor 2'. We thus obtain 

 the equation 



/,(.,i-.)= n [ r vo-r'(^ + -i)], 



the sign of multiplication extending to every combination of the values of 2, 2', 



^VO . 

 and 2Z-fl. Now let tr increase without limit, so that x=— s .. increases 



Y V "'/ 



without limit,and is of the same dimensions as e (nr (art. 125) . Observing that the 

 factor <p-X<ri)-<p- s f l<ri+ ^ l + 1 \ bas a finite ratio to e ™, if lH^, and to 



e l' aiT , if 1<-, we see that the product II <p- 8 (o-f) — f' s ( *' g, jj has a 



finite ratio to 6 t*(»)+*(»)l*» < that is to — r^- I . Hence 



^ s /" ytu+gfr X , md ^ryH-lw^ are identical, /;: denoting any term of a complete syste 

 of residues, mod y'. 



