283] DEVELOPMENTS OF THE DYNAMICAL THEORY. 301
Then we have (§ 279),
cos (a<o)
a = co -------- ,
pa
and also (§ 278),
o,j = ; cos(«®), cöz =--01-cos (ata), ... coe = —— cos (a®).
Pa Pa Pa
But from the fact that co"' is the resultant of r{" and p" we must have by
resolving along the screws of reference
= + p'"p-i, a>'"<o.i='r)"'T).2 + p"'p2,... a>"'a>e = 1)"'^ + p'"pe.(i),
whence we obtain by substitution,
ap2a2 = + p'p!, + p"'p2, ■■■ åpea6 = • • • (n).
If we multiply the first of these equations by p^lt the second by p2ß2, &c.,
and then add, we obtain
d 1p1ia1ß1 = + p'"^ßp-,
as however p is on the reciprocal system we must have, except when p'" = co ,
to be subsequently considered,
ätp12a1ß1 = -f]'"'sTß1l.
In like manner,
We shall similarly find
ß , ß ^Plßl Otj = )
>.............(in),.............
7 ^Pili “x = > 7 ^Pijißi = t"'™ß( I
whence by multiplication
///$-///>/// ///«-/'/€.'//
V ± r ^-qß^^y^^o. V Q •
But we have chosen the intensities ?/", %'", so that no one of them is
either zero or infinity, whence
rjß'^^y'^^a ~ OT?)y's5'£aOT'£/3..............(^)*
It remains to see whether this formula will continue to be satisfied in the
cases excepted from this demonstration.
Let us take the case in which pn is infinite, which makes . infinite.
We have in the case of pv very large,
/A, cos (771)
V 1 2pi
the equations (ii) become
>)"'»,cos('i;l) , „ . ^'"p c.os^pf,') „
<^Piai=----------------+P pi', ■■■ apea6---- ' +P P^
&Pi