496 
REPORT OF THE COMMITTEE ON MATHEMATICAL TABLES. 
allowed to remain, but some explanations which were given have been struck out, and 
were instead given in reference to the larger work, which is 
Reuschle, Tafeln complexer Primzahlen, welche aus Wurzeln der Einheit gebildet 
sind. Berlin, 4° (1875), pp. iii—vi and 1—671. 
This work (the mass of calculation is perfectly wonderful) relates to the roots of 
unity, the degree being any prime or composite number, as presently mentioned, having 
all the values up to and a few exceeding 100; viz. the work is in five divisions, 
relating to the cases: 
I. (pp. 1—171), degree any odd prime of the first 100, viz. 3, 5, 7, 11, 13, 17, 
19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97; 
II. (pp. 173—192), degree the power of an odd prime 9, 25, 27, 49, 81; 
III. (pp. 193—440), degree a product of two or more odd primes or their powers, 
viz. 15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 85, 87, 91, 93, 95, 99, 105; 
IV. (pp. 441—466), degree an even power of 2, viz. 4, 8, 16, 32, 64, 128; 
Y. (pp. 467—671), degree divisible by 4, viz. 12, 20, 24, 28, 36, 40, 44, 48, 52, 
56, 60, 68, 72, 76, 80, 84, 88, 92, 96, 100, 120; 
the only excluded degrees being those which are the double of an odd prime, these, 
in fact, coming under the case where the degree is the odd prime itself. 
It would be somewhat long to explain the specialities which belong to degrees 
of the forms II., III., IV., V.; and what follows refers only to Division I., degree an 
odd prime. 
For instance, if A = 7, A—1=2.3; the factors of 6 being 6, 3,“ 2, 1, there are 
accordingly four divisions, viz. I. 
I. a a prime seventh root, that is, a root of a 6 + a 5 + a 3 + a 2 + a -f 1 = 0 ; 
II. 
rj 0 = a + a -1 , 7?i = a 2 + a -2 , rj 2 = a 3 + a -3 , or 77 a root of 
V 3 + if — 2 77 — 1 = 0, 
V = 2 + Vi> V* = 2 + Vt> &c. 
■VoVi = Vo+V2> &c. 
‘VoV2 = Vi + V2, &c.; 
III. 770 = a + a 2 + a 4 , ^ = a 3 + a 5 + a 6 , or 77 a root of if -f 77 + 2 = 0; 
IV. Real numbers. 
I. p = 7m + 1. First, it gives for the several prime numbers of this form 29, 
43,.., 967 the congruence roots, mod. p; for instance, 
p 
a 
a 2 
a 3 
a 4 
a 5 
a 6 
29 
- 5 
- 4 
- 9 
- 13 
+ 7 
- 6 
43 
+ 11 
- 8 
- 2 
+ 21 
+ 16 
+ 4.