228 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
(a, b)= (a + 1) (6 + 1) ( [rm — a — b] 2 
* + [rm — a — b — l] 1 (a + b) [D] 1 
, + ab [D] 2 
J 
a (b + 1) [[rm — a — b — l] 1 "[ 
1+ bDj 
+ b (a + 1) J[rm — a — b — l] 1 
\+ ciD 
' [*]‘ 
+ ab 
M 2 , 
(a, b, c) = (a + 1) (b + 1) (c + 1) [ [rm — a — b — c ] 3 
| + [rm — a — b — c — l] 2 (a + b + c) [Z)] 1 
+ [rm — a — b — c — 2] 1 (ab + ac + be) [D] 2 
V + abc [D] 3 ) 
- [Sc (a + 1) (b + 1) [ [rm — a — b - c — l] 2 ) ] M 1 
< + [rm — a — b — c — 2] 1 (a + b) [D] 1 
l + ab [D] 2 
+ [S6c (a + 1) [ [rm — a — b — c — 2] 1 ^ ] [ac] 2 
|+ ctD J 
— abc ......... [/c] 3 . 
77. The foregoing examples are sufficient to exhibit the law; but as I shall have 
to consider the cases of four and five contacts, I will also write down the formula 
for (a, b, c, d), putting therein for shortness 
a-\~b-\-c~\~d = oi, ab + .. + cd = /3, abc.. + bed = y, abed = 8, 
a + b + c= ol, ab + ac + bc = (3', abc = y, a -\-b = a", ab = /3", a = a"; 
and also the formula for (a, b, c, d, e), putting therein in like manner 
(a, /3, y, 8, e), («', /3', y\ 8'\ (a", /3", 7 "), (a!", ¡3"'), (O 
for the combinations of (a, b, c, d, e), (a, b, c, d), (a, b, c), (a, b) and (a) respectively. 
We have 
(a, b, c, d) = 
(a + 1) (6 + 1) (c + 1) (c2 + 1) 
f [rm — a. ] 4 
+ [rm — a. — l] 3 a [D] 1 | 
- + [rm — a — 2] 2 /3 [D] 2 [ 
! + [rm — a — 3] 1 y [D] 3 J