PROFESSOR CAYLEY ON POLYZOMAL CURVES. 65 



and it hence easily appears that the equation to be verified is 



2ZT 2 «/3 - 2HL(a + B) + 2Z 2 + \ ( « -Bf . . a - B + 2(«g- Z)X 2 2(j 3H - Z) - (« - /3)x 1 

 2Z?V/3'- 2HL'(a + ff) 2L' 2 + J(«' - /30 2 " a'- /?+ 2(a'ZT- Z> 2 * 2(0 H-L') - («' -/3> 2 ° 



155. This is 



if for shortness 



.g - g . ^ + ^ + Cx 2 + Jx,Xg 



5' - (7' " " .4' + Z\ + C% + Z\x 2 ' 



A = 2 (a - B) (BH- L), A' = 2(d - B') Q3ZT- Z) , 



B = -(a-Bf , B'= -(a'-/?) 2 



G = 4 (<*£"- Z)(/3 - Z), C"= 4(«'fl"- X') (a'H - L') , 



D = - 2(« - jS) («1T- Z), D' = - 2(a' - B') (a'ZT - Z') , 



and the equation then is 



AB' - A'B + GA - GA - (X, + X 2 ) (B'G - B'G) + \\ 2 (CD' - CD - (BH - B'D)) . 



156. Calculating AB' — A'B, CA' — C'A, CD' — CD, BU — B'D, these are 

 at once seen to divide by {(a/3' — «'/3) H + L (a'— (3') —L'(a — /3')\ ; we have, 

 moreover, 



BC - B'G = -4(« - Bf (a'H - IT) (B'H - Z) + 4(«' - /3') 2 («ff - Z) (/3ZT - Z) 



= -{(aa'-/3|S')Zr-Z(a'-/S')-Z / ( a - J 8)}{(a/3'-a , /3)Zr+Z( a , -|8 , )-Z'(a-i8)}, 



viz., this also contains the same factor; and omitting it, the equation is found to be 



{(« - B) («' - /3') - 4(/3# - Z) (/3ZT - Z)} 

 -2{(««' - /3/3')ff- Z(«- /3') - Z'(« - /3)} (X 1+ X 2 ) 

 + { - (a - /S) (a - B') + 4(aZf - Z) (aZT - Z')} XjX, = ; 



viz., substituting for \ + \ and \\ their values, this is 



{(« - 0) (a' - ff) - 4(/3Zf - Z) (B'H - Z')} (if + Had - La' - L'a) 

 -{(ad - 138') H- L(d - 6')} {(a - B) (d - jS') + 4Zf if - 4XL'} 

 + { -(« - 0) («' - /?') + 4(aZT - Z) («'# - Z)} {if + fT/3/3' - LB' - L'B] = , 



which should be identically true. Multiplying by H, and writing in the form 



[(« - B) (d - B') - 4(/3ZT - Z) (B'H - L 1 )} (HM - LL' + (aH - Z) (a'fl"- Z')) 

 - {(«# - Z) (dH-L) - (BH - L) (B'H - L 1 )} ((a - B) («' - B) + ±(HM - LL')) 

 + { - ( a _ 0) («' _ £') + 4(aif - Z) (a'ZT - Z)} (Zfif - LL' + (BH - L) (B'H - L')) = 0, 



we at once see that this is so, and the theorem is thus proved, viz., that the equa- 

 tion being pp'qq' = V 2 , the foci (p = 0,p'= 0) and (q = 0, q'= 0) are concyclic. 

 157. By what precedes, X being a root of the foregoing quadric equation, we 

 may write 



qq' + 2\V + X 2 pp' = K^rr', 

 VOL. XXV. PART I. R 



