E.\r(H: 'IHF,()RV OK COMPOUND CURVES. IO7 



s-ys/rm arc all taiioi-nf to ritlu'r one or the other of 1 700 fixed circles hav- 

 ing as their centers the points P and Q and for their radii the values 



TM— TO TM f TO* 



and . 



The points J', (J. O. M. T. in fig.- 2. all lie on the same circle, hav- 

 ing the line PQ as a diameter. 



4. Among the great nuinbiT of special cases we will consider 

 the problem where the two tangents are parallel. The general 

 theorem and construction still hold, so that the solution is simpl}' a 

 matter of reduction for special values. To find the equations of 

 the locus we have to put k-^rx in formula^ (8) and (9). Observing 

 that for an intlefinitel\- large value of k 



m('_bk ! bl I Lk2^^o 

 k=--Do 



liin(^ak — a| i ^-k~ j=o 

 k=c50 



these equations become respectively 



bx— ay=o (15) 



and 



x^-j-}'- — ax — by^=o, (16) 



2 2 4 



The meaning of the equations (15) and (16) is clear: The first 

 represents a straight line through the point (a,b) and the origin; 



a b 

 the second a circle with the point ( > -) as a center, and OM as 



2.2 



a diameter, fig. 3. 



*In practical treatises on tliis subject tlic concoption of compound curves is not 

 given under tliis sfiueral point of view Thus in Mr. W. H. Scarles's treatise on Field 

 Engineering the fol lowing restriction is made: 



'"A compound curve consists of two or more consecutive circular arcs of different 

 radii, having their centers on the same side of tlie curve; but any two consecutive 

 arcs must have a common tangent at their meeting point, or their radii at this point 

 must coincide in position " 



