430 ON THE SIMULTANEOUS TRANSFORMATION OF TWO HOMOGENEOUS [74 
gfia 2 +33 1 /3 2 +GD l7 2 4-2^/3 7 +2<ffif l7 a +2^ x a/3 = n 2 gt, 
+ ^¿S' 2 + aD l7 ' 2 + 2jf a /3' 7' + 2ffi l7 ' a' 4- 2^ a' /3' = II 2 33, 
&«"* + 33 ai 8 //2 +aD l7 " 3 + 2j 1 r 7 " + 2® l7 v' + 2^ «"/3" = n 2 ©, 
&>«'<*" + 33, /3'/3" + © l7 V' + JFi W + /3V) + ©. (fee" + 7 V) 4- (o' /3" 4- a"/3') = ipjf, 
&«"« +23 1 /3"y8 +.©,/7 +^(^7 + £7")+ <£1(7"« +7«") + ^^ + a /3") = IP®, 
fUfia a' 4- 5t3i/3 /3' 4- ©i7 7' 4- Jpi (/3 7' + /3' 7) 4- (?5i (7 a' + 7'a ) + f^i (a /3' + a' /3 ) = II 2 f^, 
each of which virtually contains three equations on account of the indeterminate 
quantity A. A somewhat more elegant form may be given to these equations; thus 
the first of them is 
a, 13, 7> 
= if 
1 , 
a, Acq-t-A^ Xhi + Hi, Xgi + Gi 
. , A b 4" B, xg 4" F 
/3, Xh x 4" Hi, Xbi 4- B x , Xf 4- F x 
. , A/4- F, Ac + C 
7, Ag x 4- G x , Xf 4- F x , Xc x 4- C x 
from which the form of the whole system is sufficiently obvious. The actual values 
of the coefficients a, /3, &c. can only be obtained in the particular case where 
f 1 = g 1 = h x = F x = G x = H x = 0. If we suppose besides (which is no additional loss of 
generality) that a x = b x = c x = 1, then the whole system of formulae becomes 
(A.! 4- A) (B x + A) (C x 4- A) — H 2 
A a 4- A , A h 4- H, Xg 4- G 
Xh 4- H, A b 4- B, Xf 4- F 
Xg + G, Xf -f- F, Ac 4" C 
1 =n 2 
a, lx, g 
h, b, f 
9> f> 0 
or II 2 = /c -1 suppose ; and then 
(B 1 4- A) (Ci + A) 0? 4- (Ci 4- A) (A x 4- A) /3 2 4- (A x 4- A) (B x 4- A) 7 2 — - 
(B x 4- A) (Ci 4- A) of 2 4- (Ci 4- A) (A x 4- A) /3 2 4~ (A x 4- A) (B x 4- A) 7 - — ■- S3, 
(B x 4- A) (Ci 4" A) cc //2 4- (Ci 4- A) (Ai 4- A) /3 /2 4- (A x 4- A) (B x 4- A) 7 2 — — (&, 
(B x 4- A) (Ci 4- A) cl'a!' 4- (C x 4- A) (A x + A) 4- (A x 4- A) (B x + A) 7 , 7 // = ^ jj-, 
(B x 4- A) (Ci 4- A) cl" a 4- (C x 4- A) (A x 4* A) A ¡3 4- (A x 4- A) (B x +X^) <y"<y = — (3r, 
(B x 4- A) (Ci 4- A) a of 4* (C x 4- A) (A x 4- A) ¡3 ¡3' 4- (A x 4- A) (B x 4- A) 7 7 ,= —