49 
566] ON THE TRANSFORMATION OF AN EQUATION. 
or, as this may be written, 
X (Aa.x + .3ft + Gy x ) + Y(Aa. 2 + B/3 2 4- Gy 2 ) + %(Aa + Bfi + Gy) 
A £X 2 (a,. ft, 7i) 2 
+ 2Y~ (a,...) («2, ft, 72 ) 2 
+ ZF(c&, ...)(fli, ft, 71) (ft, ft, 72) 
+ XZ (a, ...) («i, A, 71) 0 » 0 , 7 ) 
+ YZ (a, ...) (a 2 , ft, 72) 0, £, 7 ) 
+ (a,...) (a , /3 , 7 ) 2 + &c. = 0, 
where the &c. refers to terms of the form (Z, Y, Zf and higher powers. 
But in order that the new axes may be chief axes, we must have 
Acix + Bßx + Gyx — 0, 
Aa 2 4- Bß 2 + Gy 2 = 0, 
(a, . ..)ft, ßi, 7i)(ft, ft, y2) — 9, 
so that putting for shortness 
AoL+Bß + Cy= V, 
the equation becomes 
We have 
that is, 
and thence 
WZ + ^X 2 (a,...) (ft, ft, 7i) 2 + iF 2 (a, ...)(ft, ft, 7 2 ) 2 
+ XZ {a, ...)(cLx, ft, 71) (ft ft 7) 
+ FZ (a, ...)(a 2 , ft, 7s) (a, ft 7) 
+ 1X («,...)(«, ß, y) 2 +&c. = 0. 
d. : B : C = ßx72 ft7i • 7i a 2 72 a i • a ift ®aft, 
= a : /3 : 7 , 
«, ft 7=4* 4' ft V =V(^ ! + £* + <?■). 
I write 
and also for a moment 
- = («, Ä» 7i) 2 > 
Pi 
h , g ) («!, ft, 7i), 
^ -7, / )(«!. ft, 7iX 
/ , ft, 70- 
P1/ 
C. IX. 
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