66 Proceedings of the Roijal Irish Academy. 



This proves that the sura of the pitches of three mutually rectangular 

 screws in a 3-system is constant. Of course this can be easily shown by the 

 ordinary geometrical theory of the 3-system, as given in " Treatise," p. 170. 

 I would, however, like to state that I had never noticed this theorem until it 

 recently presented itself as the natural geometrical meaning of the constancy 

 of S{ifi+j^j+ kipk). tn general wc may state that - S(i<pi+j<l)j + /><j)I;) is 

 not only the sum of the pitches of three screws that are at right angles, but 

 it is also the sum of the pitches of three screws which can be drawn through 

 a point. That this is constant is a well-known property of the 3-system.* 



Tiie fundamental theorem which expresses the relation of the system of 

 pitch-hyperboloids to the linear vector function has been given virtually by 

 Joly ;t but the following demonstration may be noted : — 



Let p be the vector from the centre of the system to some point on the 

 7>-pitcli-quadric, wliere ^j is the variable scalar expressing the pitch of a screw 

 of the system, and where if, is the linear vector function by which the system 

 is defined. 



Through any point p on the ^-pitch-quadric two generators can be drawn ; 

 and we shall suppose them parallel to vectors A and /u respectively. The first 

 of these with pitch p Ijelongs to the original 3-systeni, and the second when 

 it receives the pitch - p is a screw of the reciprocal 3-8ystem. 



The equation of the generator parallel to A is 



f, - r.^A.A-' + ./A, 



where j; is a variable scalar. 

 This may be written 



p = ^A.A' - S^A.A' + rA, 

 = ^A . A'' - ^ + .j-A, 

 or pA = ^A - 7>A + .rA', 



whence FpA = ((f,-p)X. (107) 



This is anotlier form of the equation of the generator parallel to A. 



To find the corresponding equation of the generator parallel to p which 

 Itelongs to the reciprocal system, we are to note that as the origin is now at 

 the centre, the function ^ is self-conjugate, and accordingly we have 



P = - V<pii.,r' + yp, (108) 



= -</>/'• m'' + S<pn . p-' + 7Jp, 

 " - <p^ . /J.-' + p + y^ ; 

 whence - Fpp = {<p-p)n. (109) 



As (^ - ;)) is an operator which produces a self-conjugate linear vector- 



• " Tiwitue," f. 176. t Joly, " Manual," p. 165. 



