664 



Proceedings of the Bo //a i Irish Academy. 



Since a screw of a thi-ee-systeni can be drawn parallel to any direc- 

 tion, it will be possible to make any three of the quantities 6^, . . . 6^ 

 equal to zero. Hence, we have as the condition that the three screws, 

 a, p, y, shall be all parallel to a plane the evanescence of all the de- 

 terminants of the type 



dR 



dR 



dR 



da,' 



da..' 



das 



dR 



dR 



dR 



dp,' 



dIBo 



dp. 



dR 



dR 



dR 



dy,' 



dy. 



dyz 



If the three screws, a, p, y, be co-cylindiical, these conditions must 

 of course be fulfilled; but in this case the required conditions can be 

 expressed more simply, for we must have equations of the type 



7i = Xai + fxjS,, 



76 = Aas + /x/?6, 



so that, if the three screws be co-cylindroidal, every determinant of 

 the type 



Pu 



«!, 



as, 



^2, 

 A, 



73 



must be equal to zero. 



The locus of the screws 9 perpendicular to a is represented by the 

 equation 



e.f . . . .ft'^.o. 



aa, (lag 



If we assume that the screws of reference are coreciprocal, then 

 the equation just written can only denote all the screws reciprocal to 

 the one screw whose co-ordinates are 



1 dR 

 2h da' 



1 dR 

 p^doQ 



It is manifest that all the screws perpendicular to a given line 

 cannot be reciprocal to a single screw unless the pitch of that screw be 

 infinite, otherwise the condition 



{Pa + l^d) cos ^ - d sin (;/) = 



