611] REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 497 
This means that, if a = —o (mod. 29), then a 2 =25, =.—4, a 3 =20, = — 9, &c., values 
which satisfy the congruence a 6 4- a 5 4- a 4 4- a 3 + a 2 4- a 4- 1 = 0 (mod. 29). 
Secondly, it gives, under the simple and the primary forms, the prime factors f(a) 
of these same numbers 29, 43,.., 967 ; for instance, 
p /(a) simple. /(a) primary. 
29 a -f or — a 3 2 f 3a — a 2 4- 5a 3 — 2a 4 + 4a 5 
43 a 2 + 2a 6 2a — 2a 2 + 4a 4 — a 5 — 5a 6 . 
The definition of a primary form is a form for which mod. X, 
and /'(a) = /(1) mod. (1 — a) 2 . The simple forms are also chosen so as to satisfy this 
last condition; thus /(a) = a + a 2 — a 3 , then /(1) — /(a) = 1 — a — a 2 + a 3 = (1 — a) 2 (1 + a), =0 
mod. (1 — a) 2 . 
II. p = 7 m — 1. First, it gives for the several prime numbers of this form 13, 
41,.., 937 the congruence roots, mod. p; for instance, 
p 
Vo 
Vi 
V-2 
13 
- 3 
- 6 
- 5 
41 
- 4 
4- 14 
-ii; 
and secondly, it gives, under the simple and the primary forms, the prime factors f(y) of 
these same numbers 13, 41,.., 937 ; for instance, 
p f(v) simple. f(v) primary. 
13 ?7o+2 rj., 3 + 77/j 
41 4 + rjo — 11 + 7tj l —7rj 2 . 
Thus 13 =(rjo + ^V-2)(Vi + 2?7o) (v> + 2i?i), as is easily verified; the product of first and 
second factors is = 4 + Sy« + 8^! + oy 2) and then multiplying by the third factor, the 
result is 42 + 29 (y 0 4- r). 2 ), = 13. 
III. p = 7m 4- 2 or 7m 4- 4. First, it gives for the several prime numbers of this 
form 2, 11,.., 991 the congruence roots, mod. p; for instance, 
p Vo Vi 
2 0 - 1 
11 4 -5; 
and secondly, it gives the primary prime factors f(y) of these same numbers; for instance, 
P f(v) 
2 Vo 
11 1 - 2 Vl . 
C. IX. 
63