- - ■' - '■ ■ 
14] ON LINEAR TRANSFORMATIONS. 
For example, for the orders 3, 5, 7, 9, we have 
Aio = — 2 {4 (ac — 5 2 ) (bd — c 2 ) — (ad — be) 2 }, 
Aio = - 2 {4 (ae - 45(7 + 3c 2 ) (5/- 4cc + 3c7 2 ) - (of- 3 be + 2 cd)% 
A io = - 2 {4 (ag - 65/+ 15ce - 10c7 2 ) (bh - 6cg +15 df- 10e 2 ) - (ah - bbg + 9c/- 5de) 2 }, 
Aio = — 2 {4 (af — 85A + 28c^r — 50(7/ + 35e 2 ) (bj — 8ci + 28dh — 56eg + 3of 2 ) 
— {aj — 7 bi + 20 ch — 28 dg + 14c/) 2 }. 
The derivatives D will be presently calculated in a completely expanded form up to 
the ninth order. We have, therefore, still to find the derivatives of the sixth and 
eighth orders, and a second derivative of the ninth order. For the sixth order, the 
simplest method is to make use of D 222 , which is easily seen to be equal to 
24 a, b, c, d 
b, c, d, e 
c, d, e, f 
d, e, /, g 
For the two others we have the general formulae 
A-*,*.o = 2 (A-3(A 0 A °) A-«(A 2 A 2 ) - 4A-2(A 0 A>) A- 2 (A 2 A J ) 
+ A-. (A« A 2 ) A-2 (55 2 A») + 2 [A-, (A 1 A >)] 2 }, 
2 2 2 
where U’°, A’ 1 , A 12 have been written for A’ 0 , A’ 1 , A 2 ; a formula which is demon 
strated in precisely the same way as that for 
D f _ 3.3.0 = - 2 { A- 3 (A 0 A 3 ) A_ 3 (^ 3 A 0) - 6 A-3 (A • A») B f _ z (A- 3 A- 2 ) 
+ 6А-з (A 0 A’ 2 ) A/_ 3 (A- 3 A ■ 1 ) + 9A-3 (A 1 A- 0 Б^з (A 2 A- 2 ) 
- 9A-3 ( a 1 A 2 ) A-3 ( a 2 A J ) - A-3 (A' 0 A 3 ) A_ 3 (A 3 A •)}, 
0 
(in which A’ °, &c. stand for A’ °, &c.). In particular 
D m = 2 {4 (ag — 65/ + 15cc — 10c7 2 ) (ci — 6dh + 15e^r — 10f 2 ) — 4 {ah — 5bg + 9cf— 5de) x 
{bi + bcli + Odg — oef) + (ai — 6bh + 16c^r — 2 6df+ loe 2 ) 2 + 8 (bh — бед + 1 odf— 10e 2 ) 2 }, 
D e so = — 2 {4 (ag — 65/+ 15ce — 10c7 2 ) (dj — 6ei + 15/Л — 10g 2 ) — 6 (ah — 5bg + 9 cf— 5 de) x 
(cj — 5di + 9e/i — 5fg) + 6 (ai — 6bh + 16cg — 2Gdf+ 15e 2 ) (bj — бег + 16dh — 2бед + 15/ 2 ) 
+ 36 (bh - 6cg + lodf- 10e 2 ) (ci - 6dh + 15e# - 10/ 2 ; - 9 (bi - 5ch + 9dg - be/) 2 
— (aj — 65г + 15сЛ — lddg + 9ef) 2 }. 
Hence we have all the elements necessary for the calculation of the following table 
of the independent constant derivatives of the fourth degree, up to the ninth order. [I 
have arranged the terms alphabetically and in tabular form as in my Memoirs on 
Qualities, and have corrected some inaccuracies]; 
14—2