534 Proceedings of the Royal Irish Academy. 



Let «i, «2, ^3j ^i, and hi, §2, ^3, ^i; 



be the co-ordinates of two of the comers of the tetrahedron of double 

 points in the general case ; then we have 



afli = 11 (!!i + 12fl;2 + 13^3 + 14flJ4, 



affj = 21 ffi + 22 fl2 + 23 a^, + 24 ^4, 



0^3= 31 fli + 32 ^2 + 33 (?3 + 34 ^4, 



a% = 41 «i + 42 ^2 + 43 «3 + 44 a^, 



and similar equations beginning with yli, yl^, &c. 



Multiplying each of the second system of equations by A,, and add- 

 ing to the first respectively, we have 



affi + AyJi = 1 1 (fl!i + A^i) + 12 [a^ + \l^ + 1 3 (^3 + A5j) + 14 (fl^ + XJ^), 



and three similar equations. 



Hence we learn that the point corresponding to 



«i + A^i, a^ + A52, ^3 + A&3, 04 + AJ4, 

 is 



a^i + Ay^i, aa^ + Ay^2» ct% + Ay^a, a^i + Ay54, 



so that the anharmonic ratio of these poiats and of a and h is 



Z. 

 a 



but, by what we have just seen, this must be equal to 



S 



and hence aS - ySy = 0. 



In other words, the product of one pair of the roots of the biquadratic 

 must be equal to that of the other pair. The symmetric fimction 



(a/3-yS)(ay-/?S)(aS-/3y) 



must therefore vanish, and the required theorem has been proved. 



The ;p%ich of the twist which corresponds to the displacement can 

 now be expressed in a very simple manner. 



