EXAMPLE II. 17-1=0; 16=2.8; also, the powers of 3 from 0 to 15, when divided by 17, leave remainders 1 3 9 10 13 5 15 11 16 14 8 7 4 12 2 6, ··p=a+a+a13+a15+a16+a® +a+a2 a3+a+a13+a2+a1+a® +a11+alo therefore, p and q are roots of z2+z—4=0. Next, the periods p, q may be resolved respectively into the periods therefore, r, s are roots of z2-pz-1=0; and, similarly, t, & are roots of z -qz-1=0. Lastly, the periods r, s, t, u may be resolved respectively into then r1tr2=r, .. 71, 71⁄2 are roots of z2-rz+t=0; and T1, the greatest root of this equation, =a+ä =2 cos π 17° For further information upon the theory of numbers, the student is referred to the Théorie des Nombres of Legendre, the Disquisitiones Arithmeticæ of Gauss, of which there is an excellent French translation (Recherches Arithmétiques) by Poullet-Delisle; to Barlow's Theory of Numbers; to the article of Ivory in the fourth volume of the supplement of the Encyc. Britan.; and to the Memoirs by Libri, in tome v., 1838 (Étrangères), and by Cauchy, in tome xvii., 1840, of the Mémoires of the French Academy of Sciences. 91272 0255 0556 0857 1158 1459 1760 2061 2362 2663 2964 3265 3566 3867 4168 4469 4770 5070 5371 301 5672 5973 6273 6574 6875|| 7175 7476 7777 8077 8378 11 30 2 60 8678 8979 9280 9580 9881 3 90 0181 0481 0782 1082 1383 4 120 1683 1983 2284 2584 2884 3184 3485 3785 4085 4385 5151 4685 4985 5286 5586 5886 6186 6486 6786 7086 7386 6 181 211 7686 7986 8285 8585 8885| 91835 9485 9785 8241 0084 0384 9 271 11 30 300 30 162. 163. 164. 5177 5477 5776 6075 6375 0684 0984 1283 1583 1883 2182 2482 2781 3081 3380 0264 0563 0862 2656 2955 3254 3553 3852 5644 5943 6241 6540 6839 8630 8928 9227 9525 9824 1614 1912 2210 2508 2807 0553 0851 1148 1446 1743 0068 0365 0662 1161 1460 1759 2058 2357 299 9 270 | 30 2 60 0122 0420 0719 1017 1315 4120 6179 8239 298 0255 9/209 1 30 260 389 4 119 5149 2041 2339 2636 2934 3231 0959 1256 1553 1850 2146| 2 59 6179 7 209 8 238 268 3 10 1340 1636 1932 2228 2525 208 8740 9035 9331 9627 99228238 296 1 30 9267 2 59 30 40 68 50 1470 69 167.0218 0514 0809 1105 1400 0258 2027 2322 2617 2912 3207 4975 5269 5564 5859 6153 7920 8215 8509 8803 9098 0864 1158 1452 1746 2040 0269 0563 0856 1412 1704 1997 2290 2583 86 172.0188 0480 0773 1065 1357 83 20 84 45 30 1485 40 50 10 55 9 50 10 1495 55 174. 0113 1863 2154 2446 2737 3028 4776 5068 5359 5650 5941 7688 7979 8270 8561 8852 0598 0889 1180 1471 1761 20 96 175. 30 40 50 176. 0044 0333 0623 IN THE SEVERAL DEPARTMENTS OF LITERATURE, PUBLISHED BY Harper and Brothers, New York. 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