238 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
is a conic (that is r — 2) first in terms of m, n, k, and finally in terms of m, n, a 
(a = 3m + k, as above). The results are 
for r—2, that is, curve C r a conic. 
[1]= 
2 m + 
2A- 
K 
= 
2 tn + 
?? 
= 2 ?» + 
n 
[2]= 
3m + 
6A- 
2k 
= 
3?? + 
K 
= 
a 
[3]= 
4m + 
12A- 
3 k 
= - 
ém + 
6» + 
3 k 
= - 4»? - 
3?? + 
3a 
[4] = 
5 ?•»? + 
20A- 
4 k 
= - 
10 m + 
10?? + 
6k 
= - 10??? - 
8?? + 
6a 
[5] = 
6 rm + 
30A - 
5k 
- - 
18?» + 
15» + 
10k 
= - 18?» - 
15» + 
10a 
[1,1]=- 
12 m - 
20A + 
7 k 
= - 
4??? - 
10»- 
3k 
= - 4/??- 
n - 
3a 
[1, 2]=- 
24m - 
60A + 
16k 
= 
12»? - 
30?? - 
14 k 
12 m 
12?? - 
14a 
[1,3]=- 
40?in - 
136A + 
29 k 
= 
56?» - 
68»- 
39 k 
= 56??? + 
49» - 
39a 
[1, 4]=- 
60rm - 
260A + 
46 k 
= 
140?/? - 
130?? - 
00 
= 140»? + 
122» - 
84a 
[2, 2]= - 
45 m - 
144A + 
32 k 
= 
54?» - 
72?? - 
40 k 
= 54?/? + 
48» - 
40a 
[2, 3]= - 
72rm - 
288A + 
54k 
= 
144»? - 
144» - 
90k 
= 144?» + 
126» - 
90a 
[1, 1, 1] = 
160m + 
352A- 
98k 
= - 
32»? + 
176?? + 
78k 
= - 32?» - 
58?? + 
78a 
[1, 1, 2] = 
360?•??? + 
1056A - 
240k 
= - 
336»? + 
528» + 
288k 
= - 336?» - 
336?? + 
288a 
[1. i, 3] = 
672?•??? + 
2528A - 
478k 
1184?» + 
1264?? + 
786k 
= - 1184»?- 
1094» + 
786a 
[1, 2, 2] = 
756m + 
2700A + 
530k 
__ 
1188?» + 
1350» + 
820k 
= - 1188»? - 
1110» + 
820a 
[1, 1,1, 1]=- 
3360m - 
8928A + 
2106k 
= 
2208»? - 
4464» - 
2358k 
= 2208?» + 
2610» - 
2358a 
[1, 1, 1, 2]= — 
8064m - 
26784A + 
5376k 
= 
10656»?- 
13392» - 
8016k 
= 10656»? + 
10656?? - 
8016a 
1, 1, 1, 1] = 
96768?-?» + 296448A - 
61464k 
= - 
102912»? +148224?? + 86760k 
= -10912»? - 
112056?? +86760a 
(It may be noticed as a curious circumstance that in the last column in the 
expressions of [2], [1, 2], [1, 1, 2] and [1, 1, 1, 2] respectively, the coefficients of m 
and n are in each case equal.) 
91. In the case of the conic, (1), (2), &c. are the expressions denoted in the 
former part of this Memoir by (1 ::), (2 .*.), &c., the number of points being in each 
case such as to make in all five conditions; calculating these functions by means of 
the formulae (a)=[a], &c., the comparison of the resulting values with the values 
previously obtained will show a ‘posteriori the limits within which the formulae are 
applicable; where they cease to be applicable I find the difference, and annex it as 
a correction to the formula value: I have in some cases given what seems to be the 
proper theoretical form of this difference. We have 
(1 ::) = 2 m + n; 
(2 .*.) = «5 
(3 :) = — 4m? — 3m + 3a; 
(4 •) = — 10m? — 8m + 6a ; 
(5) = - 18m? — 15m + 10a — [— 3?m + a] (= — [t]) ;