388 Proceedings of the Royal Irish Acadeimj. 



tangent line at the point whose parameter is a-o : tfo on the emanant at 



r^i of order p may be written in either of the forms 



^ ( d d \' d d \^' .. ' 



V </jri -^ d'A,^ -dx^ ^ dyj •' ^ ^'^ 

 or 



^ / (/ d\( d d\'-^.. ' 



\ dx. ^ dy.l\ V^2 ^^ dy.) ^ ^ ■^•' 



the emanant at x-^yx of order ^ has a common tangent line with the 

 emanant at x-aj^ of order n -j? + 1. 



And, similarly, as the osculating plane at the point x-^y-, on the 

 emanant of order p at ar,yi is 



■ ■^~ I d' d" d' \( d d Y- ^, ' 



or 



/ <r- /f^ d" ^ f d d y"-p , , , 



this plane osculates likewise the emanant at x^y^ of order n- p -^2. 

 Again, if both Xiyi and x^y^ vary together, 



d 



'Ux 



d X" 



is the equation of a surface which is the locus of emanants of order/?, 

 or of order w -p. In particular, the first emanants and the tangent 

 lines are curves on the developable whose cuspidal edge is the given 

 curve. 



Mixed emanants may also be considered ; but it seems to be desir- 

 able to explain, in the first instance, a notation which may be con- 

 veniently used in discussing their properties. 



