166 REPORT — 1860. 



cient of the highest power of a; in T(x) is prime top, or, which comes to the 

 same thing, that it is unity. Let F (x) == PxQ X R . . . mod p, — P, Q, R, 

 . . . being powers of different irreducible modular functions. It may then be 

 shown that F (x) == P X Q' X R' . . . , mod p m , where P', Q', R', . . . are func- 

 tions of the same order as P, Q, R, . . . , respectively congruous to them for 

 the modulus p, and deducible from them by the solution of linear congru- 

 ences only. We have thus the theorem that F (x), considered with respect 

 to the modulus p m , can always be resolved in one way and in one way only, 

 into a product of modular functions, each of which is relatively prime (for the 

 modulus/?) to all the rest, and is congruous (for the same modulus/)) to a 

 power of an irreducible function. We may therefore replace the congruence 

 F(#) = 0, modp m , by the congruences P' = 0, modjo'", Q' = 0, mod p m , 

 R'^0, mod p m , . . . But no general investigation appears to have been given 

 of the peculiarities that may be presented by a congruence of the form 

 P'^0, mod p m , in the case in which P is a power of an irreducible function 

 (mod p), and not itself such a function — a supposition which implies that the 

 discriminant of F(#) is divisible by p. If, however, P be itself an irreduci- 

 ble function, the congruence P' = 0, mod p m , gives us one and only one solu- 

 tion of the given congruence if P be linear, or, if P be not linear, it may be 

 considered as representing as many imaginary solutions as it has dimensions. 

 In particular, if we consider the case in which all the divisors P, Q, R, ... 

 are linear, we obtain the theorem : — 



"Every congruence which considered with respect to the modulus p has 

 as many incongruous solutions as it has dimensions, is also completely reso- 

 luble for the modulus j? m , having as many roots as it has dimensions, and no 

 more." 



If £=0^, modp, be a solution of the congruence F (#) = 0, mod/?, and 

 if that congruence have no other root congruous to a v the corresponding 

 solution x = a m , mod /? m , of the congruence F (r) = 0, mod /?'", may be ob- 

 tained by the solution of linear congruences only — a proposition which is in- 

 cluded in a preceding and more general observation. The process is as 

 follows: — If, in the equation 



F(a 1 + kp) = F(a l )+/>pT'(a 1 ) + ^¥"(a l ) + ..., 



we determine k by the congruence — F (a x ) + kF' (a^^O, mod /?, (which is 



always possible because the hypothesis that (.r — a,) 2 is not a divisor of 

 F (#), mod p, implies that F'(«j) is not divisible by /?*), and then put a 2 ^ 

 a 1 + kp, mod/? 2 , we have F(a 2 ) = 0, mod p 2 . Similarly, from the expansion 



F(a 2 + fy 2 )=F(a 2 )+A/? 2 F' (o a ) + ..., 



a value of k may be deduced which satisfies the congruence F(or 2 -|-#/? 2 )=0, 

 or F(a 3 )=0, mod p 3 ; and so on continually until we arrive at a congruence 

 of the form F (a my ) = 0, mod p m . But when F (x) is divisible (for the 

 modulus/?) by (x — a) 2 or a higher power of x — a, the congruence F(\r)=0, 

 mod p m , is either irresoluble or has a plurality of roots incongruous for the 

 modulus p m , but all congruous to a for the modulus p. Thus the congruence 

 (x — a) 2 +kp(x — 6)=0, mod p 2 , is irresoluble, unless a^b, mod p ; whereas 

 if that condition be satisfied, it admits of p incongruous solutions, comprised 

 in the formula #=a+^p, mod p 2 , /i=0, 1, 2, 3, . .p— ] . 



" If F (x) s (x— aj) (x), modp, where (a^ is not divisible hyp, we have F' (x) = 

 ^ (x) + (x— a,) #' (x), modp, or F' (a,) = <j> (a{), modp. 



