2 
ON CURVILINEAR COORDINATES. 
[799 
are written to denote 
dx dx dx d 2 x d~x d?x . 
dp ’ dq ’ dr ’ dp 2 ’ dp dq ’ dq 2 ’ C "’ 
and so in other cases; 
in particular, 
x ly x 2 , x 3 denote 
dx 
dp’ 
dx 
dq’ 
dx 
dr ’ 
2/u y%) y% „ 
Zi, z 2, z 3 
dy dy dy 
dp’ dq’ dr ’ 
dz dz dz 
dp’ dq’ dr' 
I. 
The minors formed with these differential coefficients are denoted by suffixed letters 
rj, £, thus 
l?i> £ 3 denote y 3 z 3 y 3 z 2 , y$Z\ — yiZ 3 , yi z -i y? z i> 
Vi, V-2> Vi >> ZoX s Z 3 X 2 , Z 3 X 1 Z\X 3 , z x x 2 z 2 x ly 
£i> £-2) Kz » x 2y?> x 3 y 2 , x 3 y 1 x 1 y 3 , x 1 y 2 x 2 y 1} 
so that, as regards these letters £, 77, the suffixes do not denote differentiations. 
The determinant 
®i, Vi, is put = L: 
X 2, 2/2 > Z 2 
x 3> y3> Z z 
and the symbols (a, b, c, f, g, h), {A, B, G, F, G, H) are defined as follows: 
a = 
xd + 
2/i 2 + 
z 2 , 
A = 
£ 2 + 
Vi + 
b = 
X.? + 
y*+ 
Z 2 , 
5 = 
yd + 
r* 2 , 
c = 
x 3 + 
yi + 
Z 3, 
G = 
& + 
V3 2 + 
f = x 2 x 3 + y.py 3 + , 
g = X 3 X x + y 3 y x + Z 3 Z\ , 
h = x x x 2 + y x y 2 + Z \ Z 2, 
F — ^2^3 + V2V3 + £>£3 . 
G — ^3^1 VzVi *b £ 3 £i, 
H=ZlZ‘2 +VlV2 + £l £2 ■ 
We have then, further, 
X\, 
yi> 
Z 1 
= L, 
a, 
h, 
g 
x 2) 
yz, 
Z 2 
h , 
b, 
f 
X 3> 
y s , 
Z 3 
g> 
f , 
c 
Zi, 
Vi, 
?1 
= L 2 , 
A, 
H, 
G 
Z2, 
V2, 
£2 
H, 
B, 
F 
£» 
Vs, 
Zz 
G, 
F, 
G 
{A, B, G, F, G, H) = (be — f 2 , ca — g 2 , ab — h 2 , gh — af , hf — bg , fg — ch ), 
L 2 (a , b, c, f, g, h ) = (BG-F 2 , GA-G 2 , AB — H 2 , GH - AF, HF-BG, FG-GH), 
which equations are at once proved, and are fundamental ones in the theory.