CURVES WHICH SATISFY GIVEN CONDITIONS. 
167 
Supp. (1,4) = *(1,4) +(m— |)(4N+20) 
-iN+fO . 
Supp. (1, 1, 3) = *(I, 1, 3)+*(2, 3)+*.2(4, 1) 
+(m-2)(2P+2Q+5J+4E) 
— 2P— Q— 2E+J' . . . 
Supp. (T, 2, 2) = *(T, 2, 2) 
+ (m— 2)(9K + 3L-|- M+2N + 0) 
+ 3L + 2M-2N-0 
(eighth equation) 
(ninth equation) 
(tenth equation) 
Supp. (1, 1, 1, 2) = *( 1, 1, 1, 2) +*(2, 2, 1) 
+(m— |)(3E+3F+6G+2D+ H+2I+ 5J) 
-2E-2F- G-iD+iH+|I-^J+2D'-J' 
(eleventh equation) 
Observe that 
G— 2E'=0, G'— 2E=0, 
and 
3G+I+8J=3G' + r + 8J', 
relations which may be used to modify the form of the last preceding result. 
Supp. (1, 1, 1, 1, 1)= *(1, 1, 1, 1, l)+*(2, 1, 1, 1) 
+ (m-f)(A+2B+4C+3D) 
— fA— |B— fC— D' .... (twelfth equation) 
130. We may in these equations introduce on the right-hand sides in place of a 
symbol such as the symbol : for example, in the fifth equation, writing 
(2, 3)=(2^T, 3) + [(2, 3)-(2 ST, 3)], 
and therefore also 
*(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 
formulse (First Memoir, Nos. 69 and 73) we readily obtain 
Referring to 
*(5) — z=0 (first equation) 
*(4, 1) =*(m+w— 6) 
=R+J' (second equation) 
