799] 
ON CURVILINEAR COORDINATES. 
3 
( £i> Vi> X 
Vi, £ 
f, g, h ][c£o, dq, dr)-, 
F, G, H^dx, dy, dz) 2 . 
h, we have = x 2 x n + yiya + z 1 z n , which 
may be written in the abbreviated form |a x = 1.11; similarly 
£ = #2^13 + y^y 13 + Z 3 Z 13 + X 3 X 12 + 2/32/ia + ¿3+2, 
which in like manner may be written £ = 2.13 + 3.12, and so in other cases; observe 
that, in the duad part of any symbol, the order of the numbers is immaterial, 
2.13=2.31. The whole system of equations is 
te = i.n 
4a, = 1.12 
4a, = 1.18 
ib 1= 2.12 
|b 3 = 2.22 
ib, = 2.23 
fa = 3.13 
^■c 2 = 3.23 , 
^•c 3 = 3.33 , 
£ = 2.13 + 3.12, 
£= 2.23 + 3.22, 
£=2.33 + 3.23, 
gl =3.11 + 1.13, 
g 2 = 3.12+ 1.23, 
g 3 = 3.13 + 1.33, 
h x = 1. 12 + 2.11, 
h 2 = 1.22 + 2.12, 
h 3 = 1.23 + 2.13. 
These may also be written 
1.11 = 4«, 
1.22= hg-ibj 
, 1.33— g 3 — ^-Cj , 
2.11 == hj "2 a 2 , 
2.22 = |-b 2 
, 2.33= £ - fa 
3.11 = gj ~ fa 
3.22= f 2 - fa 
, 3.33 = ^c 3 , 
1.23 = ^ ( — £ + g2 + h 3 ), 
1.31= ^a 3 
, 1.12 = fa 2 , 
2.23 = ^b 3 
2.31=4(f,-g s + h a ). 2.12 = 10, 
3.23 = 2^c 2 , 
3.31 =K 
, 3.12 — ^ (£ + g 2 — h 3 ). 
It is to be observed that we can, from each system of three equations, express a 
set of second differential coefficients of the x, y, z in terms of the first differential 
1—2 
It is convenient to add that we have 
f dp dp dp \ _ 1 
V dx’ dy’ dz J L 
dq dq dq 
dx ’ dy ’ dz 
dr dr dr 
dx’ dy’ dz 
that is, = 1, &c.; and, further, 
dx L 
dx 2 + dy 2 + dz 2 = (a, b , c, 
dp 2 -f- dq- + dr 2 = ~ (A, B, G, 
Differentiating the values of a, b, c, f, g,