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,

a3+a2 +a1+a2+a1+a3 +a11+a10
=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

and r1, the greatest root of this equation, =a+=2 cos a

π

17

For further information upon the theory of numbers, the student is referred to the Théorie des Nombres of Legendre, the Disquisitiones Arithmetica 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.

0 1 2 3
1440 158. 3625 3927 4228 4530 4831
6640 6941 7243 7544 7845
9653 9954

4

5

6 7 8
5133 5434 5736 6037 6338
8146 8448 8749 9050 9351

9

dif. et p.

7 22

159.

160.

10

20

15 30

25

335

3

40

50

4

40

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 1 30 8678 8979 9280 9580 9881

1683 1983 2284 2584 2884
4685 4985 5286 5586 5886
7686 7986 8285 8585 8885

0684 0984 1283 1583 1883
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 6799 7097 7394 7691
9474 9771

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

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

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

50 1475

0553 0848 1143 1438 1733
3501 3796 4091 4386 4680 295
6448 6742 7037 7331 7626|
9392 9686 9981

0275 0569
2335 2629 2923 3217 3511 5,148
5275 5569 5863 6157 6450
8213 8507 8801 9094 9388; 236.294

300

9/276

81

7

82

0269 0563 0856 2617 2911 3204 3497 3791 5551 5844 6137 6430 6723 8482 87759068 9361 9654

1150 1443 1737 2030 2324
4084 4377 4671 4964 5257
7017 7310 7603 7896 8189
9947

4118

5147

0240 0533 0826 1119
1412 1704 1997 2290 2583 2876 3168 3461 3754| 4046)
4339 4632 4924 5217 5509: 5802 6095 6387 6680 6972
7265 7557 7849 8142 8434 8727 9019 9311 9604 9896 3
86 172.0188 0480 0773 1065 1357
3110 3402 3694 3986 4278
6029 6321 6613 6905 7197
8947 9239 9530 9822

1649 1941 2233 2526 2818 5:147 4570 4862 5154 5446 5737 6176 7488 7780 8072 8364 8655 824

176.

0767 1057 1348 1638 1928
3668 3958 4248 4538 48289
204 2
6567 6857 7147 7437 77278033 3 87
9465 9754
9262 116
0044 0333 0623)

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