354 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 



Cou. 1. If C be on the opposite side of from A, the other conditions re- 

 maining the same, OC is negative. If n be even, the expression deduced in the 

 proposition remains unchanged. But if n be odd, (CPi) x (CP„) x &c., = OA" + OC". 

 And here it may be remarked, that when lines, as OA are in the original direc- 

 tion, since the coefficient of direction in that case is unity, it is immaterial whether 

 we write OA or (OA). 



Ex. Let «=3 and OC=i 



then (AC)=J,(CP,) = (CPJ=:^ 



(CA).(CP,)x(CPj=ix:^x :^=|=i + . 



Cor. 2. If C be in OA produced, the reasoning and the result will be the 

 same as in the proposition ; only, that now CA and CB being of the same affec- 

 tion, - 1 is not a divisor of the second number of the equation, and 



fCPj) X (CP„) X &c.,=(OC)''-(OA)». 



XII. If from A, the extremity of the diameter (Fig. 8), the circumference 

 be divided into n equal parts, and hues be drawn to their several extremities from 



A, then 



(AP,)x(AP,) .... (AP„_i) = « • CA-i 



As in the preceding proposition APj = CPi-CA, KP^^GP^-CA, and so on. 

 Therefore AP, x AP, x AP„_i=CP,-CAxCP^-CAx &c., to n- 1 factors 



1 



• — 



where Sj, S„, are the sum, sum of products 2 and 2, &c., of all the values of 1" except 

 unity, there being no line drawn from A to the circumference in the direction 



ir" — 1 r 



CA. S,, S„, &c., are, therefore, the coefficients of the equation — — -, or of 



^n-i _,__jn-2 +1=0, with the signs changed for the products of odd numbers 



of roots, unchanged for even ones. 



If, therefore, n-\ be even, S„_i= +1, S„_2=-l, and so on. 



If n-1 be odd, S„_i= -1, S„_o= +1, and so on. 



.-. APixAP^x&c, = .R''-ixdb{l + l + l torn terms}=±»R"-^ 



according as n-1 is even or odd. 



If ;r=T: be even, then APj + AP, x &c. =(APi) x (APJ&c, the several pairs of 

 coefficients of direction giving unity as their product. 



1 



