(IV) 



374 report — 1865. 



| L )=(— ) ( — — ). Prom (1) and (3) we infer that the series (A) is equal 

 to the series 



s/XWs' - (C) 



(in -which the summations S'S' extend only to those values of x and y for 

 which /acquires uneven values) ; from (2) and (3), considering separately 

 the two cases in which B=P, and P=Q, we infer that the coefficient of 



is the same in (P>) and (C), N representing any uneven number ; i. e. 



that the two series (A) and (B) are identical. 



Diminishing p without limit in the equation (A)=(B), and employing the 

 equations (ii) and (iii), we find immediately 



/,(P,.7,(-Q).log(T + n v 'P, = |[2-(|)],(/) 1 „ goKro)a . (o ^ ) 



a remarkable equation which connects the least solution of the Pellian equa- 

 tion with the theory of the Theta functions. 



If we suppose (as we may do) that the form (a, b, c) is reduced, so that 



«<2 a /_ , we may approximate to the values of 6'(0, w) and d'(0, J) by 

 omitting in their developments all terms after the first, and writing 



6'(0, W )X0'(O, w') = 4,r 2 e ** a + . . . 

 Substituting in (iv), we obtain the approximative equation 



y<p ) .7,(-Q).i» g (T + u v /P)=[ 2 -(|)] i(S)pj£+iv3. W 



The following examples of this formula are given by M. Kroneckcr. If 



Q=l, the exponential ■J-e** v/D is approximately equal to 2+ s/5, 18 + 5 >s/13, 

 882 + 145^/37 when we attribute to D the values 5, 13, 37; again, 

 | c tV"-'Vi7 =s4+ ^ 17j T 2_ e J ,W 9 7 =5 no4 + 5G9 v / 97, if D=17, 97; lastly, if 

 D = 85, and we give to Q. in succession the values 1, 5, 17, we find 



l e ,W- = 3 7S + 4 lv , 85) 1 e ^^ 5 =4 +Vl7 s e- WS5 =2+V5. 



These approximate representations of quadratic surds by exponentials are very 

 remarkable ; a similar observation bad, however, already been made by M. 

 Hermite *. If D=3, mod 8, the equation of which the roots are the values of 

 <j> 8 ((o) corresponding to the improperly primitive classes of det. — D, resolves 

 itself into factors of the form (at — .r+l) 3 + a(.r 2 — .v)'-=Q ; and, in particular, 



if (2, 1, — 2— j is the only such class, a is an integral number. Attributing 



to w the value "V — ' aiKl ' substituting for .r, or ^ s (w), its approximate 



value a-=2 4 (e''™-8e 2!Vw + 44e 3i '™) (equation (14) or (26), art. 124), we find 



e WD_744 

 a = 256 ' neai ' ly - 



Thus if D=43, M. Hermite has found that e* V' 3 =S8473G743. 9997775 . . ; 

 and that if D = 163, the decimal part of e v ^ l63 commences with twelve nines, 



* Tiieorie des Equations Modulalres, p. 48. 



