170 
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE 
Supp. (I, 1, 1, 2) = *(I*1, 1, 1, 2)+2*(2Z, 1, 2) 
+H+2I4-D'+ J' 
+ (m— |)(3E+3F + 6G+2D+ H+2I+ 5J) 
-2E-2F- G-iD+|H+|I-^J + 2D'-J'. 
(eleventh equation) 
Supp. (1, 1, 1, 1, 1)= *(1*1, 1, 1, 1, 1)+2*(2*1, 1, 1, 1)+D' 
+(m-f)(A+2B+4C+3D) 
— |A — |B— fC— 2D— D'. (twelfth equation) 
132. Hence finally, merely collecting the terms, we have the following expressions 
of the Supplements in the twelve equations respectively. 
Supp. (5) =N+0 (first equation) 
Supp. (4,1) =2J+2R+J' (second equation) 
Supp. (3, 2) =6K-j-4L-f-2M+3N-|-30 (third equation) 
Supp. (3, 1, 1) =D-f-E+F-j-2G-j-H + 2I+3J+D'+2J 7 (fourth equation) 
Supp. (2, 3) =*(2*1, 3) + Q (fifth equation) 
Supp. (2, 2, 1) =*(2*I, 2, 1) + 3G+I+4J+3J' (sixth equation) 
Supp. (2, 1, 1, 1) =(*2*1, 1, 1, 1)+B4-4C+4D + 2D' (seventh equation) 
Supp. (1,4) =*(1*1,4) +(4m— 7)N+(2m— 1)0 (eighth equation) 
Supp. (1, 1, 3) =*(1*1, 1, 3) +2*(2*1, 3) 
+(2m— 6)P+(2m— 5)Q+(5m— 10)J+(4m— 8)B+4J' . (ninth equation) 
Supp. (T, 2, 2) =*(1*1, 2, 2) 
+(9m— 18)K+3mL+(m+2)M + (2m— 6)N+(m— 3)0 . (tenth equation) 
Supp. (I, 1, 1, 2) =*(L d, 1, 1, 2)+2*(2a, 1, 2) 
+(2 m- 5)D+(3m- 9)E+ (3m- 9)F+(6m- 15)G 
4-(m— l)H-|-(2m— l)I+(5m— 15)J + 3D' .... (eleventh equation) 
Supp. (T, 1, 1, 1, 1 )=*(ffl, 1, 1, 1, 1)+2*(2*1, 1, 1, 1) 
+ (m— 4)A+(2m— 7)B+(4m— 12)C+(3m— 10)D, . (twelfth equation) 
where I recall the remark, ante, No. 126, that in each equation the Capitals belong to 
the system obtained by diminishing the barred number by unity and removing the 
bar; (4) for the first equation, (3, 1) for the second, and so on. 
