Multiplying the numerator and denominator by a 2 + 3, we have in the numerator a term
in 8a 3 , which may be got rid of by means of the pa-equation; the numerator for
- A„ thus becomes
{p 8 + 17p 6 + 102p 4 + 225p 2 -|-97} —P \ ^ (p 8 +14p 6 + 63p 4 + 70p 2 — 7) + 8 (p 2 + 7)
P
12 (—p 4 — 9p 2 — 9) + 16p (p 2 + 7) a
+ (a 2 — l)p
and we finally obtain
1 A _ 12 (-p 4 - 9p 2 - 9) + 16p (p 2 + 7)a + (a 2 - l)p~ 4 (p 8 + lip 6 + 37p 4 + 20p 2 + 2)
p (a 2 + 3) (p 4 + op 2 + 1)
82. The expressions obtained above for p 2 , A lt A 2 are of the form
M+Na
Pi + Qiix + RiCt
S (a 2 + 3)
1 a P% + + -B 2 a~
p 2_ £(a 2 + 3)
where
M = p 4 + 29p 2 + 49 ; A r = 8p (p 2 + 7) ; S=p 4 + 5p 2 + l,
Pi = 12 (—p 4 — p 2 + 7)-p (p 8 + I7p 6 + 102p 4 + 225p 2 + 97), Q 1 = - 16p (p 2 + 7),
jKj = p (p 8 + 17p 6 + 102p 4 + 22op 2 + 97);
P 2 =12(— p 4 -9p 2 — 9) — p~ 4 {p 8 + lip* + 37p 4 + 2Op 2 + 2), Q 2 = 16p (^> 2 4- 7),
B, = p -1 (p 8 + lip 6 4- 37p 4 + 2Op 2 4- 2) ;
substituting these values in the foregoing equation
12M 2 = 6A j 2 + 8aA 2 -p 4 + 7,
we obtain
P<i + Qa® + -ß 2 a 2 ( _ (6 (Pi + Qiß + Pja 2 ) 2 g a -Pi + Qi a + Pi a 2 (if + Pa) 2 ^ .
12 " j N (7' i :i i' ;
that is,
S 2 (a 2 + 3) 2
S (a 2 + 3)
S 2
p 12 (P 2 + Q 2 a + P 2 a 2 ) 2 (3 + a 2 ) £ ^ Pl + ^ + + 8a,S @ + ^ Pl + + ^ a2 )
- (M + Pa) 2 (3 + a 2 ) 2 + 7$ 2 (3 + a 2 ) 2 },
which, by means of the pa-equation
/ Q[3 Q/y\
p 8 + 14p 6 + 63p 4 + 7Op 2 - f—- j 8p - 7 = 0,
should be reducible to the form
p = Aa 2 + Ba + G, or p =
Aa + B
Coc + D’
but I have not been able to obtain, in either of these forms, a simple expression of
p as a function of p, a. Supposing it obtained, the pa-equation, ante No. 51, would
of course be thereby transformable into the foregoing pa-equation. And considering
p as an auxiliary parameter thus introduced into the formulae in place of p, then
/3 and the coefficients A 1} A 2 are, by what precedes, expressed in terms of p, a, that
is, in effect in terms of p, a; and we thus have the formulae of transformation for the
C. XII. 70
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