318 REPORT — 1875. 



mantissse of -n, -r, -;t . . . — :r, or, what is the same thing, the mantissse of 



pl^ pl^ ij^V- jf- 



10 lOr lOr"-^ 



Por instance, p'^=ll, 10-^1 (mod. 11), whence /=2, e=5 ; and taking 

 ^=2, we have lO^r' (mod. 11). 



The required mantissae, denoted in the table by 



(0) (1) (2) (3) (4) 



10 10.2 10.2- 10.2' 10.2* 

 are those of YV^YV ' -^T^ "IT' ^T ' 



viz. these fractions are respectively = 



(0) (1) (2) (3) (4) 



•9090 . . , 1-8181 . . , 3-6363 . . , 7-2727 . . , 14-5454 . . ; 



or their mantissse are 90, 81, 63, 27, 54. 

 And we accordingly have as a specimen 



11 (1)..81 (2).. 63 (3)... 27 (4).. 54. (0)90. 



Or again, as another specimen, r=2: — 



27 I (1) . . . 740 (2) . . 481 (3) . . 962 (4) . . 925 (5) . . 851 (0) . . 370. 



The table in this form extends to p'*=463 ; the values of r (not given in 



the body of the table) are annexed, p. 420. 



In the latter part of the table 23*^=467 to 997, we have only the mantissae 



„ 100 . 

 01 — n-. A specimen is 



1828153564 8994515539 3053016453 3820840950 

 547 6398537477 1480804387 5685557586 8372943327 

 2394881170 0182815356, 



viz. the fraction ——=-182815 has a period of 91, =4 546, figures. 



546 



Art. 3. [F. 13. The Pellian Equation.'] 



27. The Pellian equation is y-=ax^ + l, a being a given integer number, 

 which is not a square (or rather, if it be, the solution is only i/=l, x=0), 

 and a-, y being numbers to be determined: what is required is the least values 

 of X, y, since these, being known, all other values can be found. A small 

 table «=2 to 68 is given 



Euler, Op. Arith. Coll. t. i. p. 8. The Memoir is " Solutio problematum 

 Diophanteorum per numeros integros," pp. 4-10, 1732-33. The form of the 

 table is 



a >r(=:p) y(=q) 



2 2 3 



3 12 

 5 4 9 



68 4 33. 



