400 



Similarly, by transferring the origin to O", where OO"=6.OJ, the 

 equations to the circle become 



O"P=x".OI+y" . OJ, ,r" 2 + <y' + 6) 2 =:c 2 . 

 The radical loci, when y" is clinant, will have for their equations 



O"R=,r".OI + r".OJ, r" + 6=0, 

 that is, the line OjO, and 



O"S=ff". OI + *". OJ, x"*-s"*=c\ 

 that is, the rectangular hyperbola S'^, S" 2 . Hence the roots will be 



O 3 R + i.O 3 S" 1 _ O 3 R+Rm\ _ O a m\ 

 OJ OJ : ~OT 



and 



O 3 R + i.O 3 S" 2 _ O 3 R+Rm' 2 _ O 9 m\ 



oj oj : oj ; 



and the factors corresponding to them in the product (5) will be, 

 after replacing OJ by JO, as there shown, 



o,o,-oXi oo a -o,<-* f iQ, <o a 



JO JO = ~JO~'~JO~* 



Hence the final equation in (8) gives 



Oipy 1 /O 3 M f 1 O 3 M' 2 \ /O,B\ a , 



c) ' Vo 2 < ' ox;; ' (O.B) " 



Now the second factor of the product on the left-hand side of this 

 equation, is itself a product of which each factor = 1, and it may 

 consequently be omitted from the equation, so that 



OjG^OJi 



2 C - O 3 B* 



The under sign cannot be taken, because it would necessitate one of 

 the points B or C lying without, and the other within the correspond- 

 ing side of the triangle. Hence we must have 



O 1 C_O 1 B 



0~0 3 B ; 



that is, BC will be parallel to O 3 O 2 , a result which readily follows 

 from other considerations. 



12. The great complexity of the process, and the necessity of 

 knowing the algebraical form of the equation to the curve, will pro- 

 bably prevent this employment of clinant roots from having any prac- 

 tical value beyond the completion of the theoretical harmony between 

 the abstract and the concrete, the algebraical formula and the geo- 

 metrical figure. 



