520 Proceedings of the Royal Irish Academy. 



7. The order of the congruency is now determined to be three. 

 For, as p is the vector to an arbitrary point, and as cr {= p' - p) is 

 parallel to a line of the congruency through that point, the three axes 

 of <^ are parallel to the lines through the assumed point. 



8. In order to determine the class of the congruency, or the 

 number of lines which lie in an assumed plane SXp = 1, it is only 

 necessary to express that two corresponding points P and P' lie in 

 the plane. Referring to the first equations of Art. 5, if SXzs = 1 and 

 SXzs' =\, then 



xa{SXa-l)+yh{8Xli-\) + %c{8\y-\) = 0, 



and xa'iSXa'- 1) + yb'(SXfS'- 1) + zc'{SXy- 1) = 0. 



Taking the vector expression for the line PP' in terms of x, y, and 2^ 

 which is given in Art. 3, the equation of the sinyle line of the con- 

 gruency which lies in the plane is found on elimination oix, y, and z 

 to be given by the determinant, 



sa{p-a) + ta'{p-a') sh{p- /3) + ih'{p- (3') sc{p-y) 4 tc'{p-y') 

 a{SXa-l) h{SXft-l) c{8Xy-l) 



a'{SXa'-l) b'{SX/3'-l) c'{SXy'-l) 



= 0. 



As this is the equation of a single definite line, the class of the 

 congruency is unity. 



Of course when the plane is not arbitrary, more than one line 

 may lie in it. For instance, if the plane is one of the divided planes, 

 60 that SXa = SX/3 = SXy = 1, the determinant vanishes identically, 

 and no longer detennines a definite line. Indeed, this plane contains 

 an infinite number of lines joining points in it to the corresponding 

 points on the other plane which lie on their common intersection. 



9. By the general relations connecting the order and class of a 

 congruency with those of its focal surface, the focal surface of the 

 congruency under discussion is seen to be a developable of the fourth 

 degree. This appears also on consideration of the reciprocal con- 

 gruency which is of the first order and the third class, and whose 

 lines are, therefore, chords of a twisted cubic. ^ 



The equation of the focal surface may, however, be obtained with- 

 out difficulty. It is necessary to calculate, in the first instance, the 



1 Salmon, " Three Dimensions," Art. 457. 



