330 REPORT — 1875. 



I. (2) then gives f(a) the actual ideal prime factor of these same prime ^ 

 47, 139. . . .967; viz. the whole of this portion of the tahle X = 23, 1. (2) is, 

 having actual prime factors, 



599 a +a''-a''' 

 691 a' + a'' + a" 

 829 a^ + a' + a''; 



having ideal factors, their third powers actual, 



47 a' + a' + a' + a"' + a''-a-° + a'' 



139 1 -ci'-a' + a" + a'' + a'' + a'' + a'' + a'' + a'' + a-' 



277 a'-a' -a' + a' -a'° -n''-a'' + a'' + a'' 



461 a-a' + a'-a' + n"-2a'' 



967 a'-a'-a' + a"' + a'"-2a'' + a'' + a'\ 



I repeat the explanation, that for the number 47 this means f(a)f(a') .... 

 /(a-) = 47^_ 



And the like further complication presents itself in the part III. of the 

 same tahle, A = 23 (not, as it happens, in part II., nor of course in the con- 

 cluding part IV., which is a mere enumeration of real primes); thus III. (I) 

 we have congruences (mod. p^), 



p = 2, „E=-2,p = 3, Vo= +12, &c. ; 



and having actual prime factors, 



P f(v) 



69 5~2r,^ 



101 1-4^,; 



and having ideal prime factors, their third powers actual, 



p rcr,) 



3 1-2;,^; 



aa regards these last the signification being 



2^ = (l — t]^)(l — n^), T)^ + r] =—1, %r}, = G (as is at once verified), 

 3« = (1-2,„)(1-2;,J; 



but the simple numbers 2, 3 are neither of them of the form {a + bt]^)(a + hri^). 



COXTENTS OF EePORT 1 875 ON MATHEMATICAL TaBLES. 



§ 7. Tables F. Arithmological. 



Art. 1. Divisors and Prime Numbers 306 



2. Prime Roots; the Canon Arithmeticus &c 313 



3. The Pellian Equation 318 



4. Partitions 320 



5. Quadratic forms a'^-\-b'^ &c., and partitions of Numbers into Squares, Cubes, 



and Biquadi-ates 324 



0. B inary. Ternary, &c. Quadratic and higher forms 326 



7. Complex Tlieories 332 



