The Circular Locus. 



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I 



19 



Fig. III. 



.M and X, the equation of the tangent is MX + NY = r'; whence 

 it is seen that, for the point of the tani^ent at which x=0, 



r^ . 



Y will be - . Accordingly, that point may be located by 



N 



finding (in direction as well as in magnitude) a third pro- 

 portional to the quantities N and r. For this purpose, if 

 the vector ON expresses the quantity N, while OR is that 

 radius of the circle which coincides with the real axis, an 

 angle ROH must be made equal to NOR, and an angle 

 ORH equal to ONR, when the distance OH, cut off by the 

 bounding line of one of these angles on that of the other, 

 will represent the value of y. If OK make a positive angle 

 of 90" with OH, and be made equal to OH in length, it 

 will represent ?Y and hence x + iY; so that K is the point 

 of the tangent locus for which x=0. But this point must 

 be on the a:-comitant zero, and that line is accordingly QK. 

 A parallel to QK through T is the linear .r-comitant at 

 that point, and will be found to touch the j'-comitant of 

 the circular locus. Similarly, by first finding the point of 

 the tangent locus for which Y=0, the tangent ^-comitant 

 may be constructed; and in like manner both the comi- 



