20 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
calamoid along a curve (as contrasted with a solitary point) : the curve is 
then a characteristic curve. Let the neighbouring calamoid be specified by 
+ + + ^^d suppose that the line-element 
((5^c, ^t) is in the intersection, so that from (22) we must have 
— d'p^ . Sx - dq^ . 8?/ = 0 | 
- dp^ . 8x - dq^ . 8^ = 0 1 
Now by differentiating (20) and (21) we have 
— hydq-^ 4 - d^dqr, - lixdp-^ - dydp^ 4- d^dr = 0"| 
dydq-^-\-hxdq .2 4- d^dp^ - hydp^ 4- h^d7' = 0 v 
- p^dq-i^ -^V\d% + %dp^ - q^dp^^ - dr = o] 
dr from equations (23) and 
(24), we have 
= 0. 
(23) 
(24) 
j of dq^, 
dp^, 
dq^, 
djr 
8x 
hy 
0 
0 
0 
0 
0 
0 
Sx 
Sij 
— hx, 
— hy 
dz 
-dy 
dx 
dx 
dy 
hz 
hx 
^2 
-P 2 - 
- 1 
-dl 
Pi 
Performing the operations col. '4 = col. 4 — col. 3, col. '5 = col. 5+g?i col. 3, 
this gives 
= 0 . 
Sx 
hy 
0 
0 
0 
0 
0 
0 
Sx 
— hx 
— hy 
dz 
dy d\d^ d X + P\d^ 
dx 
dy 
h 
— hy dd^z hx 4 ~ pd^ z 
d2 
-P 2 
-1 
0 
0 
Expanding by Laplace’s formula in terms of minors taken from the 
three first columns and the two last columns, we have 
= 0 (25) 
Sx 
Sy 0 
Sx Sy 
- 
Sx Sy 0 
Sx Sy 
— hx 
hy dg 
— hy — qd^z hx ~ pd^z 
dx dy hg 
-dy-qplg dx + ppiz 
d2 
P 2 -1 
d2 P 2 
But if in the second of equations (21) we replace —dz by its value 
{hxdx + hydy)lhzy the equation becomes 
hxdx 1" hydy 4“ zdyq^ 4- hxti zQ 2 dx^^ zPi zP 2 Pifl 2 ^^2 2^2 — ^5 
or 
{hx + Pihz){dx 4- q^hz) + {t^y 4- q±t 7 ^){dy ~ pd^z) ~ 0 ^ 
so we can write 
— dy-\- pd^z _ dx 4~ qd^z 
hx + pd^z hy 4- qd^z 
M say . 
(26) 
and similarly 
tiy-\- pd^z _ X'^Jl'zd z _ IV 
dx^p^d, ~ dy + q^d, ~ 
. (27) 
