243-246] iioMOGiiAPHic SCREW SYSTEMS. 263 



There are in general three points, which coincide with their corre 

 spondents. These are found by putting 



& = Pi ; ft* = pa., ; j3 3 = pa a . 



Introducing these values, and eliminating a l , a. 2&amp;gt; a s , we obtain the following 

 equation for p : 



0-| (11) -p, (12), (13) 



(21), (22) -p, (23) 



(31), (32), (33) -p 



If we choose these three points of the vertices of the triangle of reference, 

 the equations relating y with x assume the simple form, 



& = /i i ; & =/ 2 &amp;lt;* 2 ; & =/ 3 a , 



where /!,/ 2 ,/ 3 are three new constants. 



245. Homographic Screw Systems. 



Given one screw a, it is easy to conceive that another screw ft correspond 

 ing thereto shall be also determined. We may, for example, suppose that 

 the co-ordinates of ft ( 34) shall be given functions of those of a. We might 

 imagine a geometrical construction by the aid of fixed lines or curves by 

 which, when an a is given, the corresponding ft shall be forthwith known : 

 again, we may imagine a connexion involving dynamical conceptions such as 

 that, when a is the seat of an impulsive wrench, ft is the instantaneous screw 

 about which the body begins to twist. 



As a moves about, so the corresponding screw ft will change its position 

 and thus two corresponding screw systems are generated. Regarding the 

 connexion between the two systems from the analytical point of view, the 

 co-ordinates of a and ft will be connected by certain equations. If it be 

 invariably true that a single screw ft corresponds to a single screw a, and that 

 conversely a single screw a corresponds to a single screw ft ; then the two 

 systems of screws are said to be homographic. 



A screw a. in the first system has one corresponding screw ft in the 

 second system ; so also to ft in the first system corresponds one screw 

 in the second system. It will generally be impossible for a and a to coincide, 

 but cases may arise in which they do coincide, and these will be discussed 

 further on. 



246. Relations among the Co-ordinates. 



From the fundamental property of two homographic screw systems the 

 co-ordinates of ft must be expressed by six equations of the type 



