14] ON LINEAR TRANSFORMATIONS, 
according as / is even or odd. Giving to / the six forms 
Qg, 6$r + l, 6g+ 2, 6g + 3, 6g + 4<, 6g + 5, 
the corresponding numbers of the independent derivatives are 
g, g, g, g + i, g, g +1; 
thus there is a single derivative for the orders 3, 5, 6, 7, 8, 10,... two for the orders 
9, 11, 12, 13, 14, 16, ... &c. 
When f is even, the terms 7)/_ 3 3 0 , Df_ 6 6 0 ..., and when f is odd, the terms 
Df- 1,1,0> Df- 4,4,0) H/—7,7,0) &c. may be taken for independent derivatives: by stopping 
immediately before that in which the second suffix exceeds the first, the right number 
of terms is always obtained. Thus, when /=9 the independent derivatives are A 
Duo, and we have the system of equations 
7) 900 + AlO 4" Dqqi — 0, 
Aio 4- T) 720 + D m = 0, 
-D720 ■t Dwo 4- Z) 621 — 0, 
D m ■+- D 62 i 4“ Dqi2 = 0, 
D^ 0 + D 531 + Duo = 0, 
which are to be reduced by 
-^900 = AoO = 0) 
It is easy to form the table 
Dq21 4“ D 53l + D 5 22 = 0, 
Duo 4~ D^sq -j- Dui = 0, 
Dui 4- 7)441 4- 7)432 = 0, 
7)522 + 7)432 + 7) 423 = 0, 
7)432 4- T)^ 4" 7)333 = 0, 
c. 
7)200 — A“> 
b 
II 
© 
b 
I 
II 
© 
A10 — ~ ^ A 2 , 
A10) 
Aw, 
A20 = A10, 
A20 = 7), 
b 
8 
II 
O 
An = 0, 
An = 0, 
7)210) 
A21 — 0) 
7)430 = A 
An = 0, 
A21 = 0, 
Aoo = A 2 > 
7)331 = 0) 
7)400 = A 2 , 
Aio= -iA 2 , 
7)222 = 0) 
Aio = -iA 2 ) 
A20 = — § 7)330 4-A 2 ) 
7)220 = i A”) 
An = § 7)330 4- A 2 > 
An = 0) 
A30) 
A21 = — F 7)330 — F A 2 ; 
A22 = f 7)330 4* f A 2 )