ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 



587 



length, have a constant product, the locus of one extremity of their 

 resultant (the other extremity being at a fixed point) is an hypei'bola. 



Multiplication of Segments. 



The simple process by which a segment is multiplied any given 

 number of times has been already alluded to. This is done in many ways 

 by using the properties of similar triangles. 



It is often required to multiply a number of segments by one multi- 

 plier, in which case the following constructions (in which the multiplier 

 is taken as K) may be used, all depending upon the property of corre- 

 spondence in two rows of points. 



(1) Take any point (fig. 8) and set ofp the segments O A, O B, O 0, 

 O D, . . . in one direction ; take any other point P at a distance of unity 

 from this line or the line produced. Draw from P a perpendicular to the 

 direction of the line meeting it in a, and take on Pa a distance P a'=K. 



Fig. 8. 



^T,*^"- *~ ~ " " 



Draw through a' a straight line parallel to D, meeting the pencil ot 



rays P 0, P A, P B, . . . in the points O', A', B', . . . 



Then 



0'A'=K . OA 



0'B'=K.OB 



0'C'=K.OC 



(2) Now suppose the point, instead of being taken at unit distance, 

 to be infinitely distant, the pencil of rays will be parallel. Along O D- 



Fig. 9. 



C- 2>' 



