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SECONDE PARTIE. — CONFÉRENCES ET COMMUNICATIONS. — SECTION III.

The general définition of the operator V for any function u of r, 8, cp is

„ du du

V = -r- Vu VG h-

d/- dO

du àcp

Vcp ;

and as R itself is such a function,

VR

But VR — 3 absolutely ; therefore

V/- = -P

Hence for any function of r, fb ?j

_ du i du i du i

dr p + dO dp dcp dp

d 0 r d<p

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

When we pass to equilateral-hyperboloidal coordinates, r changes to the hyperbolic modulus; itis no longer \J x2 +y2 z2 but \J x2— y2 — for the double sheet, and y/—x2~hy2 -f- z2 for the single sheet. The modulus is no longer the simple length, but it still dénotés what may he called the hyperbolic length; it is the multiplier which, applied to the varying hyperbolic axis, gives the length. Also 9 now dénotés the hyperbolic co-latitude; cp remains unchanged in signification.

Let R dénoté the radius-vector as before; we now hâve

R = r | cos AO i H-—— sin AG (cos<p j -+- sino k)- l

( v/— i j

dR _ d R _n dR _

â–  V/’-t- — VO -h V<p

dQ

: p Vr + r ^ V0H- ;â–  ^ Vœ. c/t) oo

VO = -V Vcp = —— .

dp T dp

r —- r —•

^ do

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. The y/— i which occurs in the expression has no directional, but an enlirely scalar, signification. As before

dR 7 dR 7n dR

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