Ball — On Homographic Screw Systems. 443 



For example, if 



+ ai, + 02, + tts, + 04, + ttg, + Ofi 



be the co-ordinates of one screw, and if 



- tti, + ao_, + tta, + ai, + a^, +05 



be the corresponding screw, then the two systems will fulfil the re- 

 quired condition. We thus have a kind of screw involution analogous 

 to what is known as the relation of involution between the rows of 

 points on the same line. 



If we add the further restriction, that the six common screws are 

 co-reciprocal, the liomography is then of a very special type. The 

 pitch of each screw, 



pya{- + . . . +2)5ai, 

 is equal to that of the corresponding screw, and the virtual coefficient, 



J9iai/3i + . . . +2)ea6(3e, 



of two screws is equal to that of the two corresponding screws. In 

 the particular case, when the virtual coefficient is zero, we see that 

 if two screws be reciprocal, so are also the two corresponding screws. 

 The angle <^ between two screws is, however, not preserved ; for, as 

 shown elsewhere, ^ 



cos (^ = SttiySi 4- % {a^/32 + OoySi) Jh2, 



when hi2 is the cosine of the angle between the two screws of refe- 

 rence, 1 and 2. Cos <^ is thus altered when the signs of a^ and^j are 

 changed. It is also evident that the perpendicular distance d between 

 the two screws is altered, for the virtual coefficient is 



{Pa + Pp) cos (^ - d sin (^. \ [ 



"We have seen that this function, as well as pa and jiP^s, remain un- 

 altered; hence, since ^ is changed, we must also have d changed. I 

 have not hitherto seen any instance in which this highly specialized 

 form of homography is presented in a physical question. 



There is a form of correspondence, very frequently of importance, 

 which must now be considered in detail. For the sake of illustration, 

 suppose a body which is at rest, and which has two degrees of free- 

 dom, be struck by any impulsive system of forces. These forces may 

 constitute a wrench of any pitch, and anywhere, yet the movement 



' Transactions of this Academy, vol. xxv. p. 306. 



