5. To complete the theory it is proper to calculate the inverse coefficients 
21, 
2 
23, 
g, 
9JÎ 
©, 
g , 
2, 
m, 
9?, 
? 
-AF--BG 
9 
X 
— 
(Ad 4 
1 
abed 3 
+ 
6 
ac 3 d 2 
— 
4 
b 3 æ 
— 
16 
W 
+ 
39 
bàd 
— 
36 
c 6 
+ 
12 
which, omitting the factor 9, is 
= 3 (ad- — 3bed + 2c 3 ) 2 — 4d? (ard 2 — (ktbed + 4ac 3 + 4b 3 d — 3b' 2 c 2 ), 
that is = 3 TT r2 — 4cZ 2 C7 ; and calculating in like manner the other coefficients, the system 
is found to be 
3X 2 -4a'U , 3XY -4>abU , 3XZ + (2ac — №) U, 3XW+ (5ad- %c) U, 
3YX -4>abU , 3F 2 -4acU , 3YZ -(lad+3bc) U, 3YW + (2bd — 6c 2 ) U, 
3ZX +(2ac-6b 2 )U, 3ZY ~(lad + 3bc)U, 3Z 2 -4bdU , 3ZW -4cdU 
3 WX + (5ad - 9be) U, 3 WY -f (2bd - 6c 2 ) U, 3 WZ - 4<cd U, , 3 W 2 - 4d 2 U 
6. Let (X, p, v, p) be any arbitrary multipliers, and write 
21, 
4?, 
©, 
2 fi, v, p), 
23, 
g, 
m 
©, 
g, 
®, 
2, 
m, 
9î, 
23