286 
SECOND MEMOIR ON THE 
[407 
Supp. (1, 4) = «(1, 4) -f (m — f) (4iV + 20) 
- ±N + 40 
(eighth equation). 
Supp. (1, 1, 3) = «(1, 1, 3) + *(2, 3) + *.2(4, 1) 
+ (m — 2) (2 P + 2 Q + 5 J + 4 R) 
-2P-Q-2B+J' 
Supp. (I, 2, 2) = *(Ï, 2, 2) 
+ (m - 2) (9Z + 3L+ M + 2N+ 0) 
+ 3L + 2M-2N-0 
Supp. (I, 1, 1, 2) = *(ï, 1, 1, 2) + *(2, 2, 1) 
+ (m-§)(3E+3F+6G + 2D+ H+2I+ 5 J) 
-2E-2F- G-%D + %H + %I-i£J+2D'-J' 
(eleventh equation). 
(ninth equation). 
(tenth equation). 
Observe that 
G - 2E' = 0, G' - 2E = 0, 30 +1 + 8/ = 30' + 7' + 8 J\ 
relations which may be used to modify the form of the last preceding result. 
Supp.(ï, 1,1,1, 1)= K{ 1, 1, 1, 1, l) + *(2, 1, 1, 1) 
+ (m - f) (A + 2B + 40+ 3D) 
— §A — §B —— D' (twelfth equation). 
130. We may in these equations introduce on the right-hand sides in place of a 
symbol such as p the symbol p/c 1: for example, in the fifth equation, writing 
(2, 3) = (2*1, 3) + [(2, 3)—(2*1, 3)], 
and therefore also 
k (2, 3) = *(2*1, 3) + * [(2, 3) — (2*1, 3)], 
the second term * [(2, 3) — (2*1, 3)] can be expressed in terms of Zeuthen’s Capitals. 
The remark applies to all the twelve equations; only as regards the first four of them, 
inasmuch as (5*1) = 0, . . (3*1, 1, 1) = 0, it is the whole original terms *(5)..*(3, 1, 1) 
which are thus expressible by means of Zeuthen’s Capitals. By the assistance of the 
formulae (First Memoir, Nos. 69 and 73) we readily obtain 
* (5) = k — 0 
* (4, 1) = * (m + n — 6) 
= R + J' 
Referring to 
(first equation). 
(second equation).