318 REPORT — 1861. 



- The numbers in the third and fourth rows are the least values of U and T 

 in the equation T" — DU^=1. The first row of numbers is the period of in- 

 tegral quotients in the development of VD: it is continued only as far as 

 the middle quotient, or the two middle quotients, after which the same quo- 

 tients recur in an inverse order. Thus, 



180=0,8,2,8,1); 

 34-01 =(18, 1, 8,2, 8, 1); 

 6377352 = (1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, ], 5, 1, 1, 1, 1, 1, 1, 5,1); 

 62809633=(9, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1). 



The numbers in the second row are the denominators of the complete 

 quotients ; i. e. taken alternately positively and negatively, they are the ex- 

 treme coefficients in the equations of the period. Thus the period of equa- 

 tions for V357is [—33, —18,1], [1,18, —33], [—33, -15,4], [4,17, 

 -17], [-17, —17,4], [4,15, —33]. The first half of the period of 

 equations for V97is [ — 16, —9,1], [1, +9, —16], [-16,-7,3], [3, 8, 

 -11], [-11, -3, 8], [8, 5, -9], [-9, -4, 9], [9, +5, -8], [-8, 

 — 3, 11], [11, 8, — 3], [ — 3, — 7, 16], the second half being composed of 

 the same equations in the same order but with their signs changed. The 

 middle coefficients of the equations are not given in the Table ; but if 

 [axj /3a., ax+i], [aA.+ i, /3a.+i, 0,^+2'] be two consecutive equations, of which 

 the former determines the integral quotient yu^, they may be successively cal- 

 culated by the formula /3a+i=/'a. "i-^+i+I^K' 



Lagrange has proved that if a-^ — Dy-=H, and H be < \/D, - is always a 



convergent to s/Y); so that a number less than ^D is or is not capable of 

 representation by the principal form of det. D, according as it is or is not in- 

 cluded among the numbers of the second row. 



The second Table of the " Canon " contains the least solution of the equa- 

 tion T^— DU^= — 1 for those values of D less than 1000 for which that equa- 

 tion is resoluble. 



Mr. Cay ley (Crelle, vol. liii. p. 369) has calculated the least solution of 

 the equation T^ — DII'=4, or T" — DTJ" = — 4, for every number D of the 

 form 8« + 5 less than 1000, for which those equations are resoluble in uneven 

 numbers. This Table, as well as Degen's second Table, is implicitly con- 

 tained in the first Table of the " Canon," as appears from the theorem of 

 Lagrange just cited. 



(ix.) The theory of the equations T-— DU^=1 and =4 is connected in a 

 remarkable manner with that of the division of the circle*. Let \=2/x + l 

 represent an uneven number divisible by k unequal primes, but having no 

 square divisor ; let also the numbers less than \ and prime to it be repre- 

 sented by a or &, according as they satisfy the equation |_|=l,or |- j= — 1 ; 

 and let X=0 be the equation of the primitive Xth roots of unity. The form of 



2a!ff 26 ITT 



this equation (seeart.59) implies that Se -^ -|-Se ^ =( — 1) ; we have also the 



2ai7r 2iiV 



relation Se ^ — Se '^ =t" >JX, which is easily deducible from the formulae of 



* See Dirichlet, " Sur la maniere de resoudre I'equation fi—pu'^=\ au moyen des fonctions 

 circulaires," Crelle, vol. xvii. p. 286. Also Jacobi's note on the division of the circle, 

 Crelle, vol. xxx. p. 173. 



