40 Proceedings of the Royal Irish Academy . 



Hence the condition 



A = W, or A/A = BjD, 



is equivalent to the condition 



(04) + (40) = 0. . . . 



If we differentiate the equations (11), we see that this is equivalent to 

 (13) + (31) = 0, (22) = ; 

 and any one of these three equations involves the other two. Differentiating 

 the three, we infer that 



(05) + (50) = (14) + (41) = (23) + (32) = 0. 

 13. There is another form in which the condition for a P-eurve may be 

 put. Let 



be the point and plane coordinates of the tangent at (x). Then all the 

 fractions Aul^ij are equal, if we suppose that i, j, k, I are 1, 2, 3, 4, taken 

 in order, so that / > i, and I > k or < k, according sls k + I is even or odd. 

 Let these fractions be equal to fx, so that 



Ajci = fi^jj, 

 then 



Ajci = /iXij + fiXij. 



Now, if the curve is a P-curve, the equations 



(00) = (22) = 

 show that the line joining the two points (x) and (.f) lies in the two planes 



(a) and (a). Hence the fractions 



ajc ai - ai a* A^i 



Xi Xj Xj Xi ^.j 

 are all equal, say to X. Hence 



AU = XXy. 



Hence 



\Xjj = jiXij + uXij, XijjXij - fi/{ij. - A). 



It follows either that all the fractions XijIXij are equal, and therefore 

 that the ratios of the Xij are constant, or else that 



H = X, /i = 0. 



The first supposition is inadmissible, and therefore, for a P-eurve, /j. is 

 constant. Conversely, if ju is constant, we have 



Aki .■= fjXij, 

 which shows that the line whose plane-coordinates are Ay is identical with 

 the line whose point-coordinates are Xij; this shows that the plane (a) 

 and the point («) are incident, and hence (22) = 0. 



