24 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



( 



_ it - /S*° 



^' 2 ' Vbd' 



Jk 



Ji 



N^p )(jU, JV, JW, JT)= 



Jp 2 



lp 2 ac r— 



be 

 fn 2 ad 



-J 



/ 2 cd 

 ab 



/?i„ad #— // 2 cd 



and 



( 



_ /Ps ab 

 V cd 



vr 4 



n/^t' " ft ' " ^i 5 ) ( yF ' 7T * ^ y? ) = ° 



1?_ 



ac 



7 



«/ 



3 > 



Jl>l 



Irn^d _ ll z hp _ ,- 

 V"bc~' * ac ' ^ 3 ' 



49. These equations may, however, be expressed in a much more elegant 

 form. Write 



a '-(/3 7 3)' V - ( 7 3a)' C ' 



c „ — '/ 



where, for shortness, {PyS) = (/3 — 7) (7 — (5) (<S — 0), &c. ; (a, 0, 7) being arbitrary 

 quantities : or, what is the same thing, 



a : b : c : d = a\p 7 d) : - V(yia) : c'(5a/3) : - d'Co/Sy) . 



Assume 



I :m : n = ga'(j3 — 7 ) 2 : ob'(y-a) 2 : rc'(a-/3) 2 ; 



7 ^2 7/ 



then the equation - f r + - = takes the form 

 a o c 



g(/3- 7 ) («-«) + <y-«) (/3-3) + r(«-/3) ( 7 -^ , 



and the four forms of the equation are found to be 



( • . M*-7)> ^(/s-«) a Jito-P)) ( ^ ' JWV ' ' Jm ' m ') =° 



*Jr(y~*), ■ , Vf (*-«), Je(a-y 



Jf(/S-7). */*(?-*)> >/r(«-£) 



viz., these are the equivalent forms of the original equation assumed to be 



(£ - 7) JpYU +(? — «) V^TF + (a - 6) J^W = 0. 



50. I remark that the theorem of the variable zomal may be obtained as a 

 transformation theorem — viz., comparing the equation y/lU + Jm V + JnW—^ 

 with the equation Jfx + Jiny + Jnz = ; this last belongs to a conic touched 

 by the three lines z = 0, ?/ = 0, z — 0; the equation of the same conic must, 

 it is clear, be expressible in a similar form by means of any other three tangents 



