A TENTH MEMOIR ON QUANTICS. 
terms of So form subdivisions such that in each subdivision the sum of the numerical 
coefficients is =0: in passing by the reversal process to the value of S 3 , the terms 
are distributed into an entirely new set of subdivisions, and then in each of these 
subdivisions the sum of the numerical coefficients is found to be = 0 ; and the like 
as regards S 1 and S 2 . 
If in the expressions for S 0 , S 1} S. 2> S 3 we first write d = e=f— 1, thus in effect 
combining the numerical coefficients for the terms which contain the same powers in 
a, b, c, we find 
S 0 = ci 3 (— 2c 3 4 6c 2 — 6c 4 2) 
4 a 2 {b 2 (6c 2 - 12c - 6) 4 b (- 15c 3 4 33c 2 - 21c + 3) 
+ 6° (42c 4 - 147c 3 4 195c 2 - 117c 4 27)} 
4 a {6 4 . 0 4 b 3 (30c 2 - 36c 4 6) 4 6 2 (- 117c 3 + 249c 2 - 1.83c 4 51) 
+ b (9c 5 + 138c 4 - 378c 3 + 330c 2 - 99c) 4 6° (- 63c 6 4 165c 5 - 147c 4 4 45c 3 )} 
4 a° {6 6 .2 4 b 5 (— 15c 4 3) 4 6 4 (75c 2 — 69c 4 24) 4 b 3 (— 9c 4 — 167c 3 4 225c 2 — 87c — 2) 
+ 6 2 (72c 5 4 48c 4 - 186c 3 4 96c 2 ) 4 b (- 126c 6 4 201c 5 - 87c 4 ) 
+ b° (27c 8 - 45c 7 + 20c 6 )} ; 
which for c = 1 becomes 
= 2¥ - 126 5 + 306 4 - Ш 3 + 306 2 -126 + 2, that is, 2 (6 - l) 6 , 
and for 6=1 becomes = 0. 
5 2 = a 3 (Oc 2 + 0c + 0) 
+ a 2 {6 2 (0c + 0) + 6 (3c 3 - 9c 2 + 9c - 3) + 6° (24c 4 - 99c 3 + 153c 2 - 105c + 27)} 
+ a (6 4 .0 + 6 3 (— 6c 2 + 12c — 6) + 6 2 (— 24c 3 + 90c 2 — 108c + 42) 
+ 6 (33c 4 - 90c 3 + 54c 2 + 30c - 27) + 6° (- 27c 6 + 78c 5 - 66c 4 + 6c 3 + 9c 2 )} 
+ a° (6 5 (3c - 3) + 6 4 (- 15c 4 15) + 6 3 (6c 3 - 12c 2 + 36c - 30) 
+ 6 2 (9c 5 — 42c 4 + 84c 3 — 108c 2 + 57c) + 6 (9c 6 — 54c 5 + 96c 4 — 51c 3 ) 
+ 6° (9c 7 - 9c 6 )} : 
which for c = 1 becomes = 0. 
5 3 = a 3 (0c + 0) 
+ a 2 {6 2 .0 + 6 (Oc 2 + 0c + 0) + 6° (18c 4 - 72c 3 + 108c 2 - 72c + 18)} 
+ a {b 3 (0c + 0) + 6 2 (- 33c 3 4 99c 2 - 99c 4 33) 4 6 (57c 4 - 162c 3 4 144c 2 - 30c - 9) 
4 6° (- 60c 5 4 207c 4 - 261c 3 4 141c 2 - 27c)} 
4 a° {b 5 .0 4 6 4 (15c 2 - 30c 4 15) 4 6 3 (- 54c 3 4 102c 2 - 42c - 6) 
4 6 2 (123c 4 - 297c 3 4 243c 2 - 87c 4 18) 4 6 (- 27c 5 4 102c 4 - 96c 3 4 21c 2 ) 
4 6° (27c 7 - 66c 6 4 51c 5 - 12c 4 )} : 
which for c = 1 becomes = 0.