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
=P+q +P +P +P +9 +9 +q=-4;

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,
r12=a11+a12+a3+a3=t,

.. 71, 71⁄2 are roots of z2-rz+t=0;
1

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
8170 8470 8769 9068 9367

0684 0984 1283 1583 1883 2182 2482 2781 3081 3380
3680 3980 4279 4578 4878
6674 6973 7273 7572 7871
9666 9965

0264 0563 0862 2656 2955 3254 3553 3852 5644 5943 6241 6540 6839 8630 8928 9227 9525 9824

1614 1912 2210 2508 2807
4596 4894 5192 5490 5788
7575 7873 8171 8469 8767

0553 0851 1148 1446 1743
3529 3826 4123 4421 4718
6502 67997097 7394 7691
9474 9771

0068 0365 0662
2443 2740 3037 3334 3631
5411 5707 6004 6301 6597
8376 8673 8969 9265) 9562

1161 1460 1759 2058 2357
4150 4449 4748 5047 5345
7137 7436 7734 8033 8331

299 9 270 | 30

2 60
3 90

0122 0420 0719 1017 1315 4120
3105 3403 3701 3999 4297 5150
6086 6384 6682 6979 7277 7 203
9064 9362 9660 9958

6179

8239 298 0255 9/209 1 30 260 389 4 119 5149

2041 2339 2636 2934 3231
5016 5313 5610 5908 6205
7988 8285 8582 8880 9177

0959 1256 1553 1850 2146|
297
3927 4224 4521 4817 5114
6894 7190 7487 7783 80801 30
9858

2 59
389
0155 0451 0747 1043 4119
5,149
2821 3117 3413 3709 4005| 6178
5781 6077 6373 6669 6965:

6179

7 209 8 238 268

3

10

1340 1636 1932 2228 2525
4301 4597 4893 5189 5485
7261 7556 7852 8148, 8444

208

8740 9035 9331 9627 99228238 296 1 30

9267

2 59

30

40 68

50

1470

69 167.0218 0514 0809 1105 1400
3173 3469 3764 4060 4355
6127 6422 6717 7012 7308
9078 9373 9668 9963

0258 2027 2322 2617 2912 3207 4975 5269 5564 5859 6153 7920 8215 8509 8803 9098

0864 1158 1452 1746 2040
3805 4099 4393 4687 4981
6744 7038 7332 7626 7920
9682 9975

0269 0563 0856
2617 2911 3204 3497 3791
5551 5844 6137 6430 6723
8482 8775 9068 9361 9654

1412 1704 1997 2290 2583
4339 4632 4924 5217 5509
7265 7557 7849 8142 8434

86 172.0188 0480 0773 1065 1357
3110 3402 3694 3986 4278
6029 6321 6613 6905 7197
8947 9239 9530 9822

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
3506 3797 4087 4378 4669
6412 6702 6993 7283 7574
9316 9606 9897
0187 0477
2218 2508 2798 3088 3378
5118 5408 5698 5988 6278
8016 8306 8596 8885 9175

20 96

175.

30

40

50

176.

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