320 



REPORT 1875. 



values of a which in Tabic I. correspond to a period of an even number of terms, 

 as shown by two equal numbers in brackets ; thus a =13, the period of ^13 

 given in Table I. is (1, 1, 1, 1) as shown by the top line 3, 1 (1, 1), and ac- 

 cordingly 13 is one of the numbers in Table II. ; and we have there 



13 



18; 



or take another specimen, 2-11 



4574225 

 71011068 



viz. the first line gives the value of x, and the second line the value of y 

 (least values), for which 'if — ax-= — 1. 



It is to be noticed that a =2 and «=5, for which we have obviously the 

 solutions (.^'=l, y=l) and {x = \, y=2) respectively, are exceptional numbers 

 not satisfying the test above referred to ; and (apparently for this reason) the 

 values in question, 2 and 5, are omitted from the table. 



30. Cayley, " Table des plus petites solutions impaires de 1' equation 

 o:-'-D7f= +4, D=5 (mod. 8)." CrcUe, t. liii. (1857), page 371 (one page). 



It is, as regards the theory of quadratic forms, important to know whether 

 for a given value of D(^o, mod. 8) there does or docs not exist a solution in 

 odd numbers of the equation, x- — D?/"=4. As remarked in the paper, " Note 

 sur I'equation x^ —'Dy-= +4, D^s5 (mod. 8)," pp. 369-371, this can be deter- 

 mined for values of D of the form in question up to D=997 by means of 

 Degen's Table; and the solutions, when they exist, of the equation a.r — D_(/'" =4, 

 as also of the equation ,r'- — D//= — 4, obtained up to the same value of D. 

 Obsei've that when the equation ,v'' — Dy"=— 4 is possible, the equation 

 a;^ — Dj/^=4 is also possible, aud that its least solution is obtained very readily 

 from that of the other equation; it is therefore sufficient to tabulate the 

 solution of x'- — J)i/'-=+4, the sign being — when the corresponding equa- 

 tion is possible, and being in other cases -|- . Hence the form of the Tabic, 

 viz. as a specimen we have 



that is, D = 757 or 781, there is no solution of either .^■-—D?/-= + 4 or = — 4; 

 D = 765, there is a solution .r=83, y = 3 of .xr^—Dy-= -{-4, but none of 

 ,r-— Dj/-= — 4 ; D = 773, there is a solution .r:L^139, y=5 of x- — J)y^= —4, 

 and therefore also a solution of x' — D(/":= +4 ; and so in other cases. 



Art. 4. [F. 14. Partitions.'] 



31. The problem of Partitions is closely connected with that of Derivations. 

 Thus if it be asked in how many ways can the number n be expressed as a 

 sum of three parts, the parts being 0, 1, 2, 3, and each part being repeat- 

 able an indefinite number of times, it is clear that n is at most =9, and that 

 for the values of in, =0, 1 ... 9 shown by the top line of the annexed table, 

 the number of partitions has the values shown by the bottom line thereof : — 



