CURVILINEAR COORDINATES, - International Mathematical Union

APPLICATION OF SPACE-ANALYSIS

TO

CURVILINEAR COORDINATES,

BY PROF. ALEXANDER MACFARLANE, Lehigh University,
South Bethlehem (Pennsylvania).

In several recent papers ( < ) , I have investigated the vector expression
for Lame's first differential parameter in the case of orthogonal systems
of curvilinear coordinates, and I have shewn how to deduce ihe expres-
sion for Lame's second differential parameter by means of direct opera-
tions of the calculus.

The results indicate that the method is not confined to orthogonal
systems, but is applicable to what may be called conjugate systems.
I shall first indicate the results for the spherical system of coordinates,
then deduce the results for the complementary system of equilateral-
hyperboloidal coordinates, and finally show how the results are modified
for an ellipsoidal system of coordinates.

Let the spherical coordinates be denoted by r, 0, cp of which r denotes
the modulus, 8 the co-latitudej and cp the longitude. If i denote the polar
axis, j and k the equatorial axes, then for any radius-vector R we have

R = r(cos0 i-t-sin0 coscpy'-t- sinO sincp k).

The axis cos 9i-f- sin 9 coscpy + sin 9 sin cpA1 may be denoted by p.
Since

R = rp,

dR = p1 dr -+- r o-t)J- c?6 -h rdc£ lcfcp.

(!) Vector differentiation (Bulletin of the Philosophical Society of Was-
hington, Vol. XIV, p. 73~92). — Differentiation in the Quaternion analysis
(Proceedings of the Royal Irish Academy, 3" series, Vol. VI, p. 199-216).

20

3o6 SECONDE PARTIE. — CONFERENCES ET COMMUNICATIONS. — SECTION III.

The general definition of the operator V for any function u of /*, 9, cp is
_ du _ du _ft du _

V = -OTr- Vr -+- -o^or VO -+- -dTo- Vc*p'I

and as R itself is such a function,

___ oK. __ (/1\ __. (/R __

=pr +r

But VR= 3 absolutely;
therefore

Vr = - V8 = -V Vcp = -V *
r^Pdb
P r^B

do

Hence for any function of r, 9,cp,

_ du £ dz* i du i
~~ dr p
dft ^ dp dy dp
rdti rdy

This operator applied to any function of r, 9, cp gives the complete
vector parameter.

When we pass to equilateral-hyperboloidal coordinates, /* changes to
the hyperbolic modulus 5 it is no longer \jx* -{-y2 -t- z2 but \Jx* — yz — z*
for the double sheet, and y/—x'2 4-y* + z'2 for the single sheet. The
modulus is no longer the simple length, but it still denotes what may he
called the hyperbolic length; it is the multiplier which, applied to the
varying hyperbolic axis, gives the length. Also 9 now denotes the hyper-
bolic co-latitude; cp remains unchanged in signification.

Let R denote the radius-vector as before; we now have

cos h 6 i H sin h 0(coscp j -+-sin cp k) !

V— i )

The expression within the brackets may be denoted by p as before; it
means the radius-vector to the surface unity and is of varying length
according to its position; but its modulus is always one. They/— i which
occurs in the expression has no directional, but an entirely scalar, signi-
fication. As before

d._R = —ddRr dr -i- -<dTmtTi-fi?6-H<d—mc dTu> ;

A. MAC FARLANE. — APPLICATION OF SPACE ANALYSIS TO CURVILINEAR COORDINATES. 807

and
_ du _ du _,. du _

__ du i du i du i
~~ dr p d6 dp d<p dp

d0 d<p

where

p = cosA8 i-\— j= sinAO (coscpy H- sines k)
V—i

•»£ = sin A8 i H—-- cos A6 (coscpy -h sincp/c)

-r£ = ••• sin/i6(— sincpy

_ sin/i6p0.
V/— i

if p0 be used to denote — sin cpy -j- cos cp A\
It is to be noted that Vr is not now the rate of change of /• per unit of

length along the normal; it is the rate of change of /• in the direction of
the radius at the point; and i per p is the actual amount. Similarly"jV 9 is
not normal to the surface 0 = constant; but has the direction of the con-

jugate axis at the point; and - per -^ is the true amount in that direction.

But V<p remains normal to the surface <p = constant, because the conju-
gate direction is identical with the normal. The vectors Vr, V9, Vconow
form a conjugate system, which is in general not orthogonal.

Return now to spherical coordinates, in order to consider the deriva-
tion of V2.

V _"_~ + ~~ +

_ A (^t l\-.

~ dr \dr p/ p

(2)

(3)

3o8 SECONDE PARTIE. — CONFERENCES ET COMMUNICATIONS. — SECTION III.

^2 ^^ , [ d . // _u_u_ i_____ \\ _i .

d /du i\ i
(6) +50V^P/_7^'

d0

d / du i \ i
("i j du I of<s> dp I OQ

\ rd^/7'd0

(8) *r
(9)
dcp

d /dM i \ i

+<M \ <>9 r *E ) <r?cip "
dO/

Suppose that the operators are applied to a scalar function of/1, 0, <p;
the only thing which remains to be determined before accomplishing the
operations is how to differentiate the reciprocal of an axis, such as p. In
the papers quoted I have shewn that an axis is differentiated in the same
manner as a scalar quantity, only the negative sign is not introduced.
Hence

term = -d3r-=*- p—2 >

i du

ri Vf _dH0y, v, ^/ , <_w

d2M i dz« i i d2,o i

' ^dcp rr ' d^c\prr

d2 u T du i dp i
" "A •"" 3 c|p ^ p25e r |

du cos0 i
d^o/'2 sin2

A. MACFARLANE. — APPLICATION OF SPACE ANALYSIS TO CURVILINEAR COORDINATES, 3og

8° term = d3c—pdrr d—p -+-d-rr rp2 d~cp d-pr-,
^ dcp dcp
qo — d*u
i I du / i P \ 2 d2pL i .
dcpd0
dp dp d0 ^ dcpd0 dp
do dcp
\ wo/ C/CP

The space-coefficient for theterms involving- j-> is

i dp i i dp i
L — -0- — -L. — •

that for the terms involving 42 ^> is

)02 dp <?P /^P\2 J.

JA \f\ P 1 ^fi I t/O

CO Ov \ O\J /

and that for the terms involvin°g —r2d-c?p-, is

i d2p j^ i cosO [
( dp \* dcp2 dp dp sin20 dp
O/CO /) wiCO 0*^(0 9
Po OI"n

[ntroduce the principles ofreduction,

(dp\2 /dp\2
d0/ ~" l (to) ~ Sm

dp dp dp do dp dp dp dp
p -(/O£ = — iwSo p' iO5o t/fC-3 = —wrC-p^('/o Or^'cPi) = — Pucp^ '

which are simply the rules of quaternious generalized; the space-coeffi-
cients then become 2, cot9, and o respectively. By application of the
same rules, the square terms become

^2' ,-2_ ^2' (rsin0)2 d-f2

and the three pairs of product terms cancel. Hence

V2 — ^1 JL— + i d^ , 2 ^ cot0d_
"" + 2 d2 ~*~ r dr "*" "r^" d0 '

Pass now to exspherical, that is equilateral-hyperboloidal coordinates.
The process of differentiation evidently remains the same as before : the

3 to SECONDE PARTIE. — CONFERENCES ET COMMUNICATIONS. — SECTION III.

question is how are the rules of reduction changed. Does p2 remain equal
to i? It does5 it means not the square of the length, but the square

of the hyperbolic length. But (-^J changes into — i, for

-\— cos/iO (coscpy -+• sincp k)

and therefore

Also ( -j- \ changes into — sin A2 9, for sin 9 changes into ~j=== sin A 9.

Because p -^ ^- remain a conjugate, although not an orthogonal system;
the other set of rules remain unchanged, viz

Hence for exspherical coordinates

l d* l

r" 5P"" (rsin/iG)2 d rdr ~~ sin/iO

Consider now the extension to ellipsoidal coordinates. The mode of
generalization will be seen bv considering the simple case of the ellipsoid
of revolution. Consider a system of similar ellipsoids, in which the ratio
of the equatorial axis to the polar axis is denoted by \. For such a system
the axis is

p = cosO -hXsin6(coscpy -h sincp A:),

where 9 denotes the elliptic co-latitude, that is, the ratio of the area of
the segment to the area of the triangle formed by the semi-axes of the
ellipse.

And

-L = — sin0 +X cosO (coscpy 4- sincp /»•),

As before -j±- = X s i n O (— sincpy -H coscp k).

and p2=I

A. MACFARLANE. — APPLICATION OF SPACE ANALYSIS TO CURVILINEAR COORDINATES. 3 [I

but

As p, -rjr and -jj-- still form a conjugate system, the other set of rules re-
main unchanged in form, that is

dp _ dp dp dp _ dp dp dp _ dp
Pd0~~~~d8p' dOd?~~~d?d0' d?P~~Pd?"

Consequently, for such a system of coordinates

_ __ du i f dw T dw i

~" dr p "^ dO dp dcp dp

r dO '' do

and

_ d2' id2 i d2 2 d _^_ cot0 d

~~ d72 + r" dO2+ r 2 X 2 s i n 2 6 d^2 + r d^ + IT2" dO "