Lanavicensis on a Theorem in Numbers. 281 



It will be observed that this number depends upon that of 

 arsenic; and I am satisfied that the number 75, as found by 

 Berzelius and Pelouze, is entitled to the utmost reliance ; the 

 respective numbers of these distinguished authorities, found by 

 entirely diflferent processes, being respectively 74-918 and 75-068. 

 My experiments serve, to some extent, to confirm Marignac's num- 

 ber, and to show that those previously obtained are much below 

 the truth. 



XLIV. On a Theorem in Numbers. By Lanavicensis*. 



TF U=0 is an algebraical equation with integer coefficients, 

 J- and V=0 another equation whose roots are ^^th powers of 

 the roots of the former {p being a prime number), it is well 

 known that V will be congruous to U quel jJ. I annex a simple 

 proof of this important proposition, which occurred to me in 

 reading over M. Lebesgue's presentation in Liouville's Journal 

 (March 1859), of Dr. Arndt's admirable proof (in which this 

 theorem plays a principal part) of the irreducibility of the equa- 

 tion to the prime roots of unity. 



To avoid complexity of notation, take an example. Suppose 



A^VB^ + Ca^^+D.r + E 

 to be U, and make jo = 5, so that U' or 



Mx'' + B'a^ + C'cr^ + D'x + E' 

 will be the product of the five quantities represented by 



Api»a;^ + B/33«x* + Cp29^f + J)pSx^ + E, 



6 being assumed successively 0, 1, 2, 3, 4, and p meaning a cer- 

 tain (arbitrarily chosen) prime 5th root of unity. Then any co- 

 efficient as C in the function so formed, will be made up of one 

 term C* and other terms which will be multiples of such argu- 

 ments as 



A*.B^CM/.E', 



subject to the conditions 



4« + 3yS + 27 + S = 2x5, 



« + /3+ 7 + 5 + € = 5 



[2 being the index to x in C*^, and 5 being the value of /> in the 

 case before us]. 



Now look at the making up of the coefficient of A".B^.C^.D*.E* 

 for any fixed system of values of a, /3, y, S, e. 



* Communicated by the Author. 



