354 BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
Cor. 1. If C be on the opposite side of O from A, the other conditions re- 
maining the same, OC is negative. Ifm be even, the expression deduced in the 
proposition remains unchanged. But if » be odd, (CP,) x (CP,) x &., =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=4 
then (AC) =3, (CP) =(CP,) =e 
(CA) . (CP,) x(CP,)=ax ‘ Veep 
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 
(OP,) x (CP,) x &e., =(O0)"—(OA)". 
XII. If from A, the extremity of the diameter (Fig. 8), the- circumference 
be divided into 2 equal parts, and lines be drawn to their several extremities from 
A, then 
(AP,)x(AP,) .... (AP,_y)=n. CA"-1 
As in the preceding proposition AP,=CP,—CA, AP,=CP,—CA, and so on. 
Therefore AP, x AP, x .... AP,_1=CP,—CA x CP,—CAx &e., to n—1 factors 
=Rr-1, { S.-1-Sn—2 Be, 8,41} 
, 1 
where §,, 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 
CA. S,, S,, &ce., are, therefore, the coefficients of the equation ae or of 
ar—l4gr-2 .... +1=0, with the signs changed for the products of odd numbers 
of roots, unchanged for even ones. 
If, therefore; »—1 be even, §,,.;=+1, S,_2=-—1, and So on: 
If n—1 be-odd, 8, ;=—1,8,_2= +1, and so on. 
AP: x AP Mes. 3 0 et RO Ay eR 1 to terms} ==tn R"-1 
according as n—1 is even or odd. 
If x—1 be even, then AP,+AP, x &c. =(AP,) x(AP,) &., the several pairs of 
coefficients of direction giving unity as their product. 
