I06 KANSAS UNIVERSITY QUARTERLY. 



x(k + l i+k2)4y — ak— b=o. (14) 



As is easily verified, the co-ordinates (lo) which represent the 

 center of one of the circles of the locus, satisfy the first of these 

 equations and those of ( 1 1 ) the second. 



If we now use the technical terminology, i. e. designate the arcs 

 MB, OB, MA, AO, etc., as arcs of compound curves and their 

 points of tangency as points of compound curves we may state the 

 theorem: 



Tlic locus of all points oj coiiipoiiiul ciii-iws be twee )i two tan^^ciiis and 

 points consists of two circles ivhich pass tliron^i:^// the two given points on 

 tzvo gi7'en tangents and wliose centers lie on the bi-sectors of the ttuo 

 given tangents. 



To construct these centers we may, therefore, connect A with B, 

 erect a perpendicular to AB in the middle of AB. which will inter- 

 sect the bi-sectors in the required points P and (J. 



Considering any point of compound curve as B, then it lies on 

 the same right line with the centers of the corresponding arcs of 

 compound curves OB and MB. Since O and B lie also on the 

 circle of the locus of points of compovmd curves with the center P, 

 the perpendicular to the chord OB througli E passes through P. 

 Hence 



<PEX.= <PDG. 



This means that every ray connecting the centers of two com- 

 pound curves whose point of tangency, or point of compound 

 curve, lies constantly on one of the circles of the locus, is tangent 

 to a fixed circle which is concentric with the circle of the locus. 

 The same can be proved for the ray EF. The two concentric 

 circles one with P, the other with O as a center, in fig. 2, are des- 

 ignated by (P ) and (Q). 



The circle of the locus with P as a center intersects the y — axis 

 and the line V in two other points J and H, such that JM=^OH, 

 and TM— TO=OH. Evidently OH is ecpial to the diameter of 

 th*^ circle (P). In a similar manner it is proved that the diameter 

 of the circle (Q) is equal to TM^-TO. To sum up we may say: 



The loens of points of compound curves of all compound curves between 

 t7vo tangents. TM and TO, and two tangent points, M and O, consists of 

 two circles which pass through the points M and O and whose centers 

 lie on the bi-sectors of the tangents TM and TO. 



By this condition, the centers P and Q of these circles and, therefore. 

 the circles themselves are perfectly determined. In all compound curves 

 the radial lines through the points of compound curves belonging to this 



