106 
ON LINEAR TRANSFORMATIONS. 
[14 
Aoo = 
A 2 , 
Aio = 
- 
2 A 2 , 
A20 = 
— 2 D 
3 -^sso 
+ 
3 A 2 ) 
An = 
2 7) 
3 -^530 
+ 
3 A j > 
B530, 
A 2 1 = 
_ 1 T) 
3 ^sso 
- 
iV A 2 ) 
£>440 = 
_ 16 T) 
15 -^530 
- 
33 A 2 ) 
£>431 = 
1 T) 
11-^530 
- 
1 7? 2 
30 -°8 ) 
£>422 = 
4 D 
T5 -^530 
+ 
2 7? 2 
13 -°8 » 
£>332 = 
1 
sh 
b 
i 
- 
1 7? 2 
T5 -°8 > 
AoO — 0, 
-AlO) 
£>720 = 
D 
81« > 
A11 = o, 
£>S30 = ^ AlO 3 -AlO) 
A2I = 2* AlO '1' 2 AlO > 
-AlO> 
-All — 2" AlO 2 -^540 ) 
A22 = 0, 
£>441 = 0, 
A32 = i £> 810 + i £> 540 ) 
£>333 = 0. 
Whatever be the value, all the tables except the three first commence thus, according 
as f is even or odd, 
£>/, 0,0 — 
A 2 > 
or A, 0,0 =0, 
11 
0 
-w, 
£>/—1,1,0) 
£>/—2,2,0 — 
— 3 A-3.3,0+3 A 2 » 
A“ 2 , 2, 0 = — A- 
£>/—2,1,1 = 
3 A-3,3,0+3 A 2 > 
£>/—2, 1,1 = 6, 
£>/—3,3,0 
but beyond this I am not acquainted with the law. 
To give some formulae for the transformation of these derivatives; we have, for 
example, 
A-1,1.0 = (12.84)^ 18. 42 UUUU = 13.42 A-. (U, U)B f _ x {U, U). 
But 13.42 = - Z&VsV* ~ + ViZ&V*, 
and WA (A U)B f . x {U, A) = A-i(№ vU)B f _ x ( V U, ZU) 
= B f _ x (A 0 A J ) B f _ 1 (A 1 A«), &c. 
1 1 
(where A °, A 1 stand for A 0 A x , &c.); or 
A-1.1,0 = - 2 (A-i (A 0 A °) A_! (A 1 A >) - A-i (A- 0 A) B f _ 1 (A 1 A•)}, 
which reduces itself to 
A-i,i,o = -2{£ / - 1 (A°A 1 )}», 
A-1,1.0 — 2 {A-1 (A» A•) A-i (A 1 A 1 ) - [A-1 (A■ 0 A 0?}, 
according as / is even or odd.