ON CURVILINEAR COORDINATES.
-
4 D
(4
+ g23
C12
- K)
+
2 L
[(f 3
-c 2 )
Xi +
(g* - Cl ) L
! + (h +
g 2 - 2h 3 ) L 3 ]
+
A
{-
cal2
+
af31 + 3.
. ag23
}
+
B
r_
bcl2
- 3.
.bf31 -
bg23
}
+
G
i
•
cf31 -
cg23
}
+
F
1
bc31
+
cfl2
•
- %23}
+
G
{-
ca23
-
cgl2 -
- fg31}
+
H
f
1
+
af23 -
bg31
i
1
[799
№
It would be possible in these equations to introduce the symbols AB12, &c., in
place of abl.2, &c., and then writing A = B = G = 1, all these symbols other than those
where the letters are GH, HF or FG would vanish, and we should obtain Mr Warren’s
six equations for normal coordinates. But in the general case it would seem that there
is not any advantage in the introduction of the new symbols AB\2, &c., and I retain
by preference the equations in the form in which I have given them.
To the foregoing may be joined a symmetrical equation obtained (as by Mr Warren)
by multiplying the several equations by a, b, c, f, g, h respectively, and adding; the
result is in the first instance obtained in the form
- 2Z 2 © + 2- 3L 2 il + □ = 0 («W),
© = a (b 33 + c 22 - 24)
+ b (c u + a 33 - 2g 31 )
■P c (a^ -P b n 2h 12 )
+ 2f (g 12 + h 31 - a 23 - f u )
+ 2g (has + 4 - b 31 - gaa)
+ 2h (4 + go3 c 12 h ; . ;j ).
For 'P, collecting the terms which contain L u L 3 , L z respectively, and attending
to the values of {A, B, G, F, G, H),
= (bc —f 2 , ca —g 2 , ab — h 2 , gh —af, hf—bg, fg —ch),
this is easily reduced to the form
' V P = (Mj + H 2 + G 3 ) L 1 + (H 1 + B 2 + F s ) L 2 + (G 1 + F. 2 + G 3 ) L 3 .
The term in il is
= - (Ma + Bh + Cc + 2Ff+ 2Gg + 2Hh) H,
= — 3L 2 £l, as above.
