394 



Proceedings of the Royal Irish Academy. 



21. Extension of a theorem of Cremona) s. 



The extension of Cremona's theorem, referred to in the note Jto 

 Art. 16, is as follows : the locus of the pole of the points in which a 

 variable first emanant meets a fixed plane, is a conic section. Or still 

 more generally, let the first emanant be 



d 



d 



'^Tx^y^Ty]"^ 



Xj, 



d d\.' 



— +1/. — f 



dx^'^d^j 



f 



and consider the locus of the pole of the points determined by the first 

 emanant 



dF dF 



^'^^^'^ = '' 



where F{xy) = is a scalar binary of the n'^ degree. In the notation 

 of Art. 6, the pole is 



d d 



^ dx ^ dy 



--D-L. 



dx 



l'^i^y^ty)^\'^rx-'y^^)^ 



,id<i> dF 



~(d4y dF\ [d4 dF\ -| Jd^ dF 

 \dx ' dy )„_i \dy' dx j„_y J ^Kdy'dy 



dF\ 



/n-\ 



'df dF\ 



^df dF\ 

 \dx ' dy /,^i ' \dy 



Jdfd_F\ -\...JdfdF\ 



'^hh'^. 



dy 'dx j,^i 



/ /7fK /7 7P\ 



Here, as in the article cited, ( 7- • ;^ ) is the (12)""^ invariant of 



\dx dx /,ij 



the binaries of the order n - 1, and it is evident, that if ^1 : yi varies, 



the locus of the poles is a conic section. 



22. Vm'cursal curve reyarded as the locus of the mean centre of corre- 

 spondiny points on any tiumber of homoyraphically divided lines. 



The general unicursal curve admits of a simple geometrical con- 

 struction. Let e-iCz . . . f-n be the roots of f{t, 1) = 0, and let the 

 curve be 



^{t, 1) (a^ai . . . a„) (^, 1)" 



P = 



f{t,\) {a,a,...a„){t, If 



