188 THE ROYAL SOCIETY OF CANADA 



(18) am''-\-2^lm-\-yP = 0, {am + l3iy = P, 

 i3m2+27/m+ 5/2 = 0, il3m-\-yiy = Pn. 



It can now be easily shown that 



A-hSBt+SCt'+Df^= -(am + /3/) itl!^ 



and 



(19) lm''A = {l-mty(an-\-ôU) = {I -mi)m^K, 

 where 



(20) m''K = {l-mt) {an-^blt). 

 Substituting in (9) and (11) we obtain 



(21) X2P2=-4/, 



(22) K^Bz = -20(aw + /3/). 



Differentiating the last equation we obtain, after reduction, 

 niK^dz' =60((^m + i3/) i0m-yl-2Ôlt), 

 w2KW= -120(aw + ,8/) [2{l3m-yl-2ôlty-\-ôlmK\. 



The invariant ^8 may now be calculated from (14). We find 



(23) K^es= 2* • 3'^ • 5- • 7 l{ani + 0l)\ 

 Hence 



(24) d\_ 3^-7^ P _Z^-V 



We are thus led to the theorem: Equations (1) will represent an 

 anharmonic cubic if, and only if, 



(25) (a6-/37) — 4(/32-a7) (7— /3Ô)=0. 



5. Singular Points. Every rational cubic has one and only 

 one double point. We propose now to investigate the character of 

 this double point when the cubic is anharmonic. The flex parame- 

 ters are the three roots of the cubic* 



(26) A=a+3iS/ + 37f' + ô/'' = 0, 



and the two nodal parameters are the roots of the quadratic* 



35, 37, 3^ 



(27) 37, 3/3, 3a =0. 



/, -t, f 



Equation (27) reduces, by means of (17), to 



(28) . {mt-iy = Q. 



*Salmon's Higher Plane Curves, third edition, pages 187-188. 



