360 Journal of the Asiatic Society of Bengal. [July, 1908. 
Thus let «= = pe and let btany 
when #= a®, we have, #°=a, 
rs 
If paat+y!, we have (2 ~y?)b=a 
or #5 ~ 6x" V y+ 152% . y — 20a5y VW y+ 15a%y? — bay’/y +y3 =a. 
Transposing the terms not containing y to the right-hand 
side and squaring we get an equation rational in z, y. If we write 
1 a 2 
b® for y, this equation is reduced to the form f,+f,.b3+f,. be = 0 
which is easily rationalized. 
10. The equations rationalized by Cayley are only particular 
cases of equations (8) of Art, 9 and can be obtained from them by 
putting #=0. 
Thus if < rationalize #= \/at/b+ fc by the method of 
Art. 9 we get 
2° — Apa’ + 2x4 (3p? —4q) — 40? (p> —4pq + 16r) + (p? — 4g)? =0......(5) 
(where p=at+b+c; g=be+catab and r=abc). 
*, Ifa=0, ¢.e., ifa/at f Vb+/Se=0; we must have p?—4q=0 
or a? +B? 408 — 2be ~ 2ea — 2ab = 
Again, if we rationalize 
a=ari/at / b+ / ct / d by the same method we get 
{x9 + 408(7d —p) + 2a*(35d? — 30pd + 3p* — 4q) + 4u°(7d3 — 15 pd? 
+ 9p'*d — 129d — p' + 40g — 16r ) + d§ — 4d + 2d2(3p* — 4q) — 4d(p — 
4pq + 16r) + p*— 8p’q + 16q?}* — 640d {a8 + &4(7d —3p) + #°(7d? — 10pd 
+ 3p* —4q) + d’ —3pd* + 3p*d — p® — 16gr+4pq—-16r}? = 0......... (6) 
If / at /b+/cr+ r/ d=, we must have 
— 4pd® + 2d?(3p? ~ 4g) — 4d (p? — dq + 16r) + (p? —4q)? =0.......(7) 
[where p, g, r have the same significations as in (5)] 
or (p?—4q + d®?—2pd)? = 64rd = 64abed, 
t.e., (a2+ b? +08 + d? —2ab — 2ac — 2ad —2be — 2bd — 2cd)* = G4abed, 
Observe that (5) reduces to (7) if we write therein d for 2%, 
similarly if we write e for 2? in (6) we can get the result of the 
rationalization of Jat / b+/ c+ d +r/ex0, The rea- 
son is obvious because Sat b4S ct dt 2 =0 
reduces to the form #=4/ @ tVbtV/ cid ifa=—v/e. 

