TRANSFORMATION OF TWO HOMOGENEOUS FUNCTIONS, &C. 
429 
74] 
provided that we have 
' aj = a«- + ba' 2 4- cot! 1 - + 2fa!a" + 2ga"a 4- 2hota' , 
b, = a/3 2 + b/3'- + c/3" 2 + 2f/3'/3" + 2g/3"/3 + 2h/3/3', 
Cj = ay 2 + by' 2 4- C7" 2 + 2fy'y // + 2gy"y + 2hyy / , 
f> = a/3y4- b/3'y' + c/3V + f (/3'y" + /3V) 4- g (/3"y + /87") + h (#/ + /3'y), 
gj = aya 4- by'a' + cy"a" + f (y'a" + y"a') + g (y''« + y 2") 4- h (yen' + y' a), 
, hj = aa/3 4- ba'/3' + ca"/3" + f (a'/3" + «"/3') + g (a"/3 + a/3") + h (a/3' + a'/3). 
Representing for a moment the equations between the pairs of functions of the 
second order by 
u = u 1 , U=Uy, v = v lt 
we have, whatever be the value of A, 
Xu -f- TJ 4- v = XUy 4- Ui 4- Vi. 
Hence, if 
« , £ , y 
o', /3', 7' 
/3", y" 
= n 
Acq + + aj, Xhy + H 1 + h x , Xg x + Gy 4~ gi 
A hj + Hy + h 1} A by 4~ Fy + b 1; Afj + Fy 4- f x 
tyfi + G x 4- gi, A/i + Fy 4- f x , ACy + (72 + 0! 
Hence, since a, b, c, f, g, h, are arbitrary, 
= n; then 
An A -f- a, A h 4~ H q- h, A// 4- (x4~g 
Ah 4~ H 4~ h, Ab 4* F 4- b, 4" F4- f 
\g + G + g, Xf + F+î, Xc+C+c 
X(iy 4- Ay, Xhy 4~ Hy, Ag x 4- Gy 
= ll 2 
An 4“ A, 
X h 4- H, 
Xg 4~ G 
Xhy 4~ Hy, Xby 4- Fy , X J'y 4~ Fy 
X h 4- H, 
X b 4* F, 
Xf + F 
Xgy 4- Gy, Xfy 4- Fy, ACj 4- Cy 
Xg 4- G, 
Xf 4- F, 
X c + C 
which determine the relations which must subsist between the coefficients of the 
functions of the second order. We derive 
a 1} 
hy, 
9i 
= № 
a, 
h, 
9 
hy, 
by, 
fi 
h, 
b, 
f 
9i> 
A> 
Cy 
9> 
/, 
c 
and by comparing the coefficients of a, &c., if we write for shortness, 
» = 
. , Xb + F, Xf+F 
. , Xf + F, Ac 4- C 
&c., then