100 



rroceedings of the Royal Irish Academy. 



The radius of inversion being unity, the point P {ic, y, z) and its inverse P, 

 (a'j, i/i, 2,) are connected by the equations 



.(• y z 



xx = -, yi = -, z, = -, 

 r 'r y- 



where r- = a;^ + y^ + s", the origin being the centre of inversion. 



If X, y, z moves along a curve of the TT-faniily deiiued by 



Idx + mdy + nd.z = 



where I- + ra?- + w = 1, it is easUy seen that *,, ?/„ j, moves along the family 

 111 defined by 



where 



and 



lidxi + niidyx + «] dzi = 



1x 2v '>z 



h=l -P, iih = ill --^P, '/I, = « - — P, 



r r- r- 



ds,^ 



P = Ix + my + «vS ; also /,' + m^ + n^^ = 1. 



Using the equations before employed, the normal toision f -) on Hi, for 



the direction dx^, dy^ ''-i inverse to the direction dx, dy, dz, is given by 



dxi dy^ dz, 



h 'Mi «l = A]. 



dli dill I diij 



Now, dl, = (// - 2Pdx, - 2.r,dP, etc. 



The terms 2P(Z'-i, 2/%,, 2J'di, disappear in A„ and x,,yuz, are expressed 

 in terms of x, y, z. 



If we border the determinant r'^, liy an upper row l,a-, ?/, -, and by a 

 left-liand column 1, 0, 0, 0, and add ob\ious multiples of the upper row from 

 each of the remaining three rows, we find 



1 



2rdr 



r-A, = 



2P 

 2dP 



X 

 dr 



y 

 dy 



dz 



I 



■III 



n 



U 



dm 



dn 



■ Now 

 therefore 



P = /.'• + '///// + 7/s, and Idx + mdy + ndz = 0, 

 dP = xdl + ydin + zdn. 



2x 2y 2z 

 Multiplying the second, thiid, and fourth columns l»y — , — , — 



