TRANSACTIONS OF SECTION A. 



473 



less th&np, be a multiple of p 2 ? " This was answered in the affirmative by Jacobi on 

 p. 301 of the same volume, who showed that 3 ,0 =1 (mod. II 2 ), 14 28 = 1 (mod. 29 2 ), 

 and 18 36 = 1 (mod. 37 2 ). (See also Jacobi's ' Canon Arithmeticus ' (1839), p. xxxiv). 

 Desmarest, in fact, obtained a solution of the congruence a?- l =l (mod. p") for 

 x = 10, viz. 10 486 = 1 (mod. 487 2 ). The fact that only one value of p should occur up 

 to 1000 for which the congruence is satisfied is not remarkable when the diminu- 

 tion of the probability of a number being a factor of its period, as we ascend in the 

 series of primes, is considered. Regarding the period merely as a number taken at 

 random, we can see how small is the chance that a large prime should leave a zero 

 remainder when divided into its period ; but there is no reason to suppose that 

 there are not values of p for which lO? -1 = 1 (mod. p 3 ), &c. 



17. Elementary Demonstration of the Theorem of Multiplication of Determi- 

 nants. By M. Falk, Docens of Mathematics in the University of TJpsala. 



The present article is intended to give a rigorous demonstration of the im- 

 portant theorem above mentioned, founded upon the same elementary principles 

 (elimination between two systems of equations) as the demonstration which is 

 given by Brioschi in his excellent Treatise on Determinants, and is reproduced by 

 Schellbach in his German translation of this work. To the demonstration of 

 Brioschi, I think one must object that it is incomplete, as taking no account of the 

 numerators of the two quotients, from the equality of which that of the denomi- 

 nators is concluded. 



In the following we use the notations : 



(1) • 

 whence 



(2) • 

 and 



(3) 



{(uv) = u 1 v 1 + u 2 v 2 + 

 (uv) 1 = u. z v 2 +u 3 v 3 + 



+ u n v,,, 



(uv) 1 = (uv) — u 1 v 1 , 



A = 



(aa), (a/3), (ay),.. .(ok) 

 (ba), (bB), (by),...(b<) 

 (ca), (c/3), (cy),...(«) 



(ka), (A/3), (ky),...{kK) 



D = 



«,, a.„ «, 



...a„ 



*1, "2, "v 



b v K h> •••*» 



It 2, 3* "••^•n 



k u k 2) k 3 , ...k n 



A = 



(4) 



(6/3),, 6y) 1 ...(6ic) 1 

 (<0)i, (<7i) — («)i 



D l = 



K K 



k 2 ,k 3 .. 



(*0)i, (*y)i 

 A. 



02, 03" A 



7» y s —y» 



Now comparing the systems (3) and (4), we see that the constituents of A in 

 (3) are composed of those of D and A, exactly in the same manner as the con- 

 stituents in Aj of those in D, and A r The determinants in (3) are of the w th , 

 those in (4) of the (n — 1)" order. The theorem, therefore, will be generally 

 demonstrated, if we prove : 



1) That it holds for determinants of the second order, and, 



2) That, if it holds for determinants of the (n — 1)", it must also be true for 

 those of the ra th order; i.e. that if A 1 =D 1 A 1 , then A = DA. 



