660 
PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
S 0 = a 3 (— 2c 3 +6c 3 — 6c+2) 
+a 3 {5 3 (6c 3 — 12c— 6)+5(— 15c 3 +33c 2 — 21c+3) 
+ 5 0 (42c 4 -147c 3 -bl95c 3 -ll7c + 27)} 
+a [5 4 0 + 5 3 (30c 2 — 36c + 6)+5 2 (— ll7c 3 + 249c 3 — 183c + 51) 
+ 6(9c 5 +138c 4 — 378c 8 + 330c 2 — 99c) +5°(-63c c ’+ 165c 5 — 147c 4 +45c 3 )} 
+a°.{6 6 . 2 + 5 6 (— 15c+3)+5 4 (75c 2 — 69c + 24) + 5 3 ( — 9c 4 — 167c 3 -f225c 3 — 87c— 2) 
+ 5 3 (72c 5 +48c 4 — 186c 3 +96c 3 )+5(— 126c 6 +201c 5 — 87c 4 ) 
+ 5°(27c 8 -45c 7 +20c 6 )} 
which for c = 1 becomes 
= 25 6 — 12& 5 + 30& 4 — 405 3 +305 2 — 126 + 2, that is 2(5-l) 6 . 
and for 5=1, becomes =0. 
So— <x 3 (0c 2 +0c+0) 
+a 2 {6 2 (0c+0)+6(3c 3 -9c 3 +9c— 3)+6°(24c 4 — 99c 3 +153c 2 — 105c+27)} 
+ci {¥. 0 + 5 3 (-6c 3 + 12c— 6) + 5 2 ( — 24c 3 + 90c 3 -108c + 42) 
+ 5(33c 4 — 90c 3 +54c 2 +30c— 27)+6°.(— 27 c 6 +78c s — 66c 4 +6c 3 +9c 3 )} 
+a°{5 5 (3c — 3)+5 4 (-15c+15) + 5 3 (6c 3 -12c 2 +36c — 30) 
+5 3 (9c 6 — 42c 4 +84c 3 — 108c 3 +57c) + 5(9c 6 — 54c 5 +96c 4 — 51c 3 ) 
+5°(9c 7 -9c 6 )} 
which for c=l becomes =0. 
S 3 = a 3 (0c + 0) 
+a 2 {6 3 . 0 + 6(0c 3 +0c+0)+5°(18c 4 — 72c 3 +108c 3 — 72c+18)} 
+ct {5 3 (0c + 0)-f6 a ( — 33c 3 + 99c 2 — 99c + 33)+5(57c 4 — 162c 3 +144c 3 — 30c — 9) 
+ 5°( — 60c 5 + 207c 4 — 261c 3 +141c 2 — 27c)} 
+rt°{6 5 . 0+5 4 (15c 2 — 30c+15)+5 3 (— 54c 3 + 102c 3 — 42c — 6) 
+ 5 3 (123c 4 — 297c 3 -h243c 2 —87c+ 18) +5(-27c 6 + 102c 4 — 96c 3 4-21c 3 ) 
+ b°( 2 7c 7 — 6 6c° + 5 1c 5 — 1 2c 4 ) } 
which for c=l becomes =0. 
