246 ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
where as before the (. .)’s refer to the like functions with the two sets of letters 
interchanged. Developing and collecting, this is 
(f> (x + oc) — (f)X — (fix' = a multiplied into 
8x\x + 8xx'~ 
+ tf 2 .6Z 2 Z' + 6ZZ /2 + 2Z' 3 
- 28ZZ' - 14Z' 2 
+ 28Z' 
+ seat. 4Z 3 + 12Z 2 Z' + 12ZZ' 2 + 4Z' 3 
- 28Z 2 - 56XX' - 28X'- 
+ 56Z + 56Z' 
-22 
whence 
+ x' 2 . 2Z 3 + 6Z 2 Z' + 6ZZ' 2 
- 14Z 2 - 28ZZ' 
4- 28Z 
+ X. - 3()Z 2 Z' - 30ZZ' 2 - 10Z' 3 
4- 140ZZ' + 70Z' 2 
- 116Z' - 6£' 
+ x . - 10Z 3 - 30Z 2 Z' - 30ZZ /2 
+ 70Z 2 + 140ZZ' 
-116Z-6£ 
+ 36Z 2 Z' + 36ZZ' 2 
- 152ZZ' 
-4(Zr + Z'|); 
cf>x= ci multiplied into 
«*( + 1) 
+ a 2 ( 2Z 3 -14Z 2 + 28Z-11) 
+ x (- 10Z 3 + 70Z 2 - 116Z + l) 
4- 12Z 3 — 76Z 2 +LX 
4- £ (— 6x — 4Z + X), 
where the constants l, L, X have to be determined. We should have cf>x = 0 for a 
cubic curve; viz. x = S: X = 6, £ = 18; Z = 4, £=12; or Z = 3, £=1(). Writing first 
x = 3, the equation is 
8Z 2 - 96Z - 72 - £ (18 + 4Z) + 3/ + XL + £\ = 0,