B18 



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189/. The limacon of Pascal.* The limagon may be defined 

 as generated from a circle by adding a r>ns:ant length t> 



each of the radii vectores 

 drawn fnmi a point on its 

 circumference as origin, 

 proper account being tak 1 1 

 of negative radii vectores. f 

 E.g., let OLA^be agiv n 

 x circle of radius a, any 

 point on it, A^ = k any 

 constant ; then if any 

 radius vector as OP l be 

 drawn from 0, and J\l* 

 = A^A = k be addrd i,, 

 it, then P is a point on the limagon ; and as P 1 is made to 

 describe a circle, P will trace the limac.on. 



The polar equation of the curve is at once written down 

 from this definition ; for, if the diameter OCX be taken as 

 initial line, then the polar equation of the circle is 



p = 2 a cos 6, . . . ( 1 ) 

 whence the polar equation of the limagon is 



p = 2acos0-f k. . . . (L') 



If A: be taken equal to a, the radius of the given cir< !<-, 

 tins equation may be written in the more common form 



This curve was invented and named by Blaise Pascal (1023-160-2), 

 celebrated French geometrician and philosopher. It is, however, a .-; 

 case of the so-called Cartesian ovals. 



t Th i limac.on may also be defined as the locus of the intersection of th 

 two lines OP and CP which are so related during tin -ir revolution about 

 and C, respectively, that the angle XCP is always equal to | times the 

 XOP. This definition easily leads to the polar equation already derived. 



