406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
233 
la, b, c, <?] = —(M + l)(& + l)(c + l)(Æ+l)(a + l)(a + 2)(a + 3) 
-(<* + l)(6 + l)( c +l)(d+l) 
.. rrn 
#0 + 1)0 + 2) (a + 3) 
-/3 0 + 2) (a + 3) 
j ~ y 0 4" 3) 
[-23 
d (a + 1) (b + 1) (c + 1) (d + 1) (a + 2) (a + 3) ^ 
cd(c + d+ 2) (a-f 1) (b + 1) (a + 3) 
+ 22 3cd (6 + c 4* d 4" 3) (a 4* 1) 
— 6 abed 
where a = a + b+c + d, ..8 = abed. 
. I>, b, c, d, e]= (a 4-1)(6 + 1)(c4-1) (c? 4 1)(e + 1) (a 4 1)(a + 2)(a + 3)(a + 4) 
4- (a + 1) (b + 1) (c + 1) (d + 1) (e + 1) 
+ 2 
- 2 
' a 
(a 4-1) (a 4-2) (a + 3) 0 + 4) 
- /3 
0 + 2)0 + 3) 0+4) 
“ 7 
o + 3) 0 + 4) 
— 23 
0 + 4) 
„ — 6e 
rm 
A 
+ ( — 0 + 1) (b + 1) (c + 1) (d 4-1) (e + 1) (a + 2) (a 4- 3) (a 4- 4) "j k, 
+ 2c?e (d + e+ 2)(a + l)(& + 1) (c + 1) (a+3)(a + 4)j 
- 2lede (c 4 d 4 e 4 3) (a 4 1) (b 4-1) 
+ dXbcde (b + c + d + e + 4) (a + 1) 
— 24abode 
where a = a + b + c + d+e, /3 = &c.,... e = abode. 
(«4-4) 
83. The complete functions (a), (a, b), (a, b, c), &c. may be expressed by means of 
the linear terms [a], [a, 6], [a, b, c], &c. as follows, viz. we have 
(a) = [a], 
(a, b) = [a] [6] 
+ [a, b], 
(a, b, c) = [a] [6] [c] 
4- [a] [b, c] + [6] [a, c] + [c] [a, 6] 
+ [a, 6, c], 
(a, 6, c, d) = [a] [3] [c] [d] 
+ 2[a][3][c, d] 
+ 2 [a, 3][c, d] 
4- 2 [a] [b, c, d] 
+ [a, b, c, d\ 
C. VI. 
30