328 REPORT — 1861. 



lines, which all meet the curve, and which, commencing with the asymptote 

 w= — - — jX, continually recede from it, and approximate to the asym- 



ptote y =— 7f^ — T ^- The sectorial area contained between any two conse- 

 cutive lines of this pencil and either branch of the hyperbola is constant and 

 equal to- . —7=- log^^ ; as may be ascertained by employing polar 



V n 



coordinates. Since the same observations apply to the pencil y=—r x, we 



infer that the lines of these two pencils lie alternately, unless the two pencils 

 coincide. Let us now suppose that in the form (a, 6, c), a is positive and c 

 negative ; so that the axis of x does, and the axis of y does not cut the curve. 



On this supposition the values of -~ and of -^ continually increase from 



— . ^ — to — - — as n increases from — oo to + co . The alternate po- 



VD + b VD— 6 ' 



sition of the lines of the two pencils gives, in this case, the theorem, — 

 •* The inequalities 



~r — — 1 — ' 



Xk a-„ — .^A + l 



in which k represents any given number, are satisfied for one value of ti, and 

 one only." If, taking a for M' and [1, 0] for [.r'(„y„], we put k=0, we 

 obtain the conclusion, — 



" Each set of representations of the positive number M by the form 

 (a, b, c), in which a is positive and c negative, contains one and only one 

 representation which satisfies the inequalities 



.r»>0, y«>0, J/„ ^^r^^ ^"'" 



It is in this form that the theorem appears in Dirichlet's memoir. We 

 may add that any values of x and y which satisfy these inequalities will give 

 a positive value to (a, h, c) ; for such a pair of values will correspond to a 

 point situated in the internal angle between the asymptotes of the hyperbola. 



The fifth section contains the demonstration of the theorem, that if A de- 

 note the absolute value of D, and i^(2A) be the number of numbers less than 

 2A and prime to it, a properly primitive form of determinant D will acquire 

 a value prime to 2D, if its indeterrainates x and y satisfy any one of a certain 

 set of 2A4'(2A) congruential conditions included among the 4A- conditions 

 represented by the formulae 



.r=a, mod 2A ; y=/3, mod 2 A, 



in which both a and /3 represent any term of a complete system of residues, 

 mod 2 A ; but will acquire a value not prime to 2D, if j- and y satisfy any of 

 the other congruential conditions. 



If the form be improperly primitive, the number of congruential conditions 

 that will render its value unevenly even and prime to A will be Ai^(A), or 

 3A\/.(A), according as D ss 1 , or eees 5, mod 8. 



These theorems are easily demonstrated by considering separately the 

 prime divisors of A. For example, if the form (a, b, c) be improperly 

 primitive, and /? be a prime divisor of D, since either a or c is prime to p, 

 let a be prime to p\ then {ax + byy—Dy'^ will be prime to jo, when 

 ax + by is so ; i. e. it will be prime top, ior p(p—\) combinations of the 



