BISHOP TERROT ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 353 



Also CAj = (CD,) + \/^ . (D, Aj) 



CA„_l = (CD,)-^/3T. (D,A,) for (D,AO = (D,A„_i) 

 CA, X CA„_i = (CD,2) + (D,AJ- which .■. = r^=iCAy 

 or its equivalent in area (CAj)^. 



XI. Cotes' Properties of the Circle. 



Let the circumference of the circle be divided into n equal parts ; and to the 

 extremities of these let lines be drawn from the 



Pig. 8. 



centre (Fig. 8), as OP,, 0P„, &c., and from any- 

 other point C in the diameter. Then 



CP, = OP, - OC, CP, = OP, - 00, &c. 



CP, X CP, X CP3 CP„ 



= 2„ . (0Ar-2„_i(0A)''-i .... ±0C" 

 Where 2,, is the product of all the coefficients 

 of direction for OP,, 0P„ &c., 2„_i, the sum of these 

 coeflBcients taken n—l together, and so on. But 



these coefficients (Prop. III.) are also the values of 

 i_ 

 1" , or the roots of the equation «" — 1 = 0. Now the 



product of the roots of this equation with their signs changed is -1, and 2, is the 



product with the signs unchanged. 



Therefore if n be even, 2„=: -1, and, ifn be odd, 2„= + 1 ; and in either case, 

 2„_i, 2„_2, &c., each =0. 



Hence CP, x CP, x CP„= ± (OA)" ± (OC)" ; the upper signs being used 



when n is even, the lower when n is odd. 



4 



But CP, CP„ &c., represent the lines considered in relation both to length 

 and direction ; therefore, to change the equation into one in which the length 

 only of these lines shall be expressed, we must divide the first side, or multiply 

 the second by the product of all their coeflftcients of direction. 



If n be even, the several pairs, as CP,, CP„_i, are evidently of the form 



A _A 



(CP,) . 1^' and (CP_i) . 1 -'' .-. CP, X CP„_i = (CP,) x (CP„_i) 



i and the same is true for every pair except CA=:(CA) . +1 and CB=(CB) . -1 



(CP,) X (CPJ CP„ = [-OA" + OC"] X -1 = 0A"-0C". 



If, again, n be odd, the several pairs remain as before, only, no P falling upon 

 B,-l is not a coefficient of direction: 



(CP, X (CP„) X &c., =OA"-OC" as before. 



VOL. XVI. PART III. 4 U 



