468 Proceedings of the Royal Irish Academy. 



6. The inflexional tangents at P are obtained as follows : — 

 Let one of them meet W in a point whose coordinates are 



Xx jxx Xy fxy 



ax" - 62 ax'' - e^ dy^ - 6^ If' - Qx ' ' 

 tlien a point on the line joining this to P will lie on the Hessian, if 



0. • 



1 + P 715 — F + P — . — ^J 



ax^ — 61 



Now, Q = 0, and -3— = are both satisfied hj p = 0, and so also 

 dp 



1 [X {ax-" -6{) + IX (ax^ - e^)^ 

 ^m {ax" - e,)\ax' - e^f ~ ' 



or, making use of the identity (5), 



thus showing that the inflexional tangents at P or i" divide harmo- 

 nically the line VV. 

 The expressions 



X 



admit of representation in another form. If % — - — - be reduced to 



ax~ - 6 



a single numerator and denominator, we get 



^ X Re (^^ -^Pd+ Q) 



ax^ - e {ax'' - 0) {hf- - 6) {cz^ - 6) {dv' - Q) {ew- - 



=f{0){e-6,){e-d,), 



where 0^ and O^ are roots of 6'' + P6 ^ Q ^ ; 



