"PROP. II.

on spherical

The function Q, which has been investigated in the pre- Digression ceding proposition, is the same as the coefficient of e' in the harmonics. expansion of the quantity

{1—2e. (1—2t)+e2 } −1.

"Let u be a quantity which satisfies the equation

(c)........ut+e.u(1-u);

Murphy's analysis.

"But if, as before, we write t' for 1-t, we have, by Lagrange's theorem, applied to the equation (e)

"If we differentiate, and put for di() its value 1.2.3...iQ; given

"by the former proposition, we get

dti

"Second Expansion.—If u and v are functions of any variable t, "then the theorem of Leibnitz gives the identity

di(uv) d'u, dv di-u, i(i-1) d'r di-v
dt di-1 1.2 dt di- +ete.

+

"Put ut and r=t', and dividing by 1.2.3...i, we have

Expansions of zonal harmonic.

"Third Expansion.-Put 1-2t=p, and therefore it' =

and tesseral harmonics.

The t, t' and μ of Murphy's notation are related to the we have used, thus :—

Also it is convenient to recall from App. B. (v′), (38), (40), and (42), that the value of Q; [or 9" of App. B. (61)], when 6=0 is unity, and that it is related to the ", of our notation for polar harmonic elements, thus:

as is proved also by comparing (g) with App. B. (38). We add
the following formula, manifest from (38), which shows a deriva-
tion of from ", valuable if only as proving that the i-s
roots of "=0 are all real and unequal, inasmuch as App. B.
(p) proves that the i roots of "=0 are all real and unequal :—
1 dr (1)
¿—s+1 dμ sin3-10]

sin 0

From this and (3) we find

=

(0)

1.2.3...(i-s)

1.3.5...(2-1)"

sins de Qi

And lastly, referring to App. B. (w); let

14

Q'i and Qi[coscos 0'+sin @sin 'cos ($—$ )] denote respectively what Q becomes when cos is replaced b cos', and again by cos@cos + sin @sin 'cos (-): and kta denote cos; and p', cos'. By what precedes, we may put 61

See Errata.

of App. B. into the following much more convenient form, agree- Biaxal ing with that given by Murphy (Electricity, p. 24):

Qi[cos cos 0'+sin @sin 'cos (-')]

dQdQ cos 2 (+-')
+
du du (i-1)i(i+1) (i+2)"

‚d2 Q1 d2Q'

sin20 sin20

harmonic expanded.

i(i+1) du2 du etc.} (6). 783. Elementary polar harmonics become, in an extreme case of spherical harmonic analysis, the proper harmonics for the treatment, by either polar or rectilineal rectangular coordinates, of problems in which we have a plane, or two parallel planes, instead of a spherical surface, or two concen- Physical tric spherical surfaces, thus:

problems relative to plane rect

circular

First, let S be any surface harmonic of order i, and V; and angular and V-- the solid harmonics [App. B. (b)] equal to it on the plates. spherical surface of radius a: so that

and, therefore, if a be infinite, and r—a a finite quantity denoted
by x, which makes

Supposing now S; to be a polar harmonic element, and consider-
ing, as Green did in his celebrated Essay on Electricity, an area
sensibly plane round either pole, or considering any sensibly plane
portion far removed from each pole, it is interesting and instruc-
tive to examine how the formula [App. B. (36)...(40), (61), (65);
and § 782, (e), (f), (g)] wear down to the proper plane polar
or rectangular formula. This we may safely leave to the ana-
lytical student. In Vol. II. the plane polar solution will be fully.
examined. At present we merely remark that, in rectangular
surface co-ordinates (y, x) in the spherical surface become plane,
S may be any function whatever fulfilling the equation.

and that the rectangular solution into which the elementary spherical harmonic wears down, for sensibly plane pertine spherical surface far removed from the poles, is

784. The following tables and graphic represent vis all the polar harmonic elements of the 6th and 7th e be useful in promoting an intelligent comprehens subject.

Tesseral,

=

(33μ* — 18μ3 + 1)

=

16