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A PEOOF BY ELEMENTAEY METHODS, WITHOUT COM- 
PLEX QUANTITIES, THAT EVEEY ALGEBEAIC FUNCTION 
(WITH EEAL COEFFICIENTS) HAS FACTOES OF THE 
FOEM {x--:px + q) {p, q, EEAL) AND HENCE, EVEEY 
ALGEBEAIC EQUATION WITH COEFFICIENTS EEAL OE 
IMAGINAEY, HAS EEAL OE IMAGINAEY EOOTS EQUAL 
IN NUMBEE TO THE DEGEEE OF THE EQUATION. 
By Professor W. N. Eoseveare, M.A. 
(Communicated by Professor Lawrence Crawford.) 
(Eead June 17, 1914.) 
Let \aoX"-{-aj^x''~' -\- ... +a^^\, = 2c, represent any algebraical function of x, 
of degree n, having its coefficients real : we proceed to consider whether it 
has a factor of the form (x^ -px + q), where ^ are real. 
§ 1. The remainder when 2C is divided by {x^-px + q) is, as is well 
known, obtained by putting x^ - px + q=0, i.e. x^=px-q, whence it 
follows that x' =Pi_^x - q^ ; 
qi being determined by x'+' =p._^x^ — q.x 
=Pi_,{p>x — q) — qiX. 
Whence 
Pi = VPi-.-qi and qu.^qPi-A 
• •• Pi=ppi-.-qPi-. J ^ 
This reduction-formula combined with p)^^ 1, q^ = 0 (i.e. p_-, = 0), gives 
all the values of Pi and g^. 
In the above work i may be a negative integer ; we now proceed to 
show that the jj's and q's for negative integers are closely connected with 
those for + integers. 
Thus if 
x'' =p_i_,x — qp_i_2 
and 
x^ =Pi_iX — qp^_,_, 
