In this section we study the set of points {(£, r)) | ^(5, n) = 0} , which we call the curve V , and then give the explicit form of the
spectral representation of I(x, y) for all mass configurations
allowed by the stability conditions. The function F(£, r\) is defined in eq. (4-12); it is a quadratic polynomial in £ and r\ . One
factorisation of F(z,, n ) is given in eq. (4-12), the functions /+(£) being given by eq. (4-11) or, alternatively, by
/+(?) = (S2-l) Nce-lKad+ioWa+fcHc+cO
±[(5-€1+) (5-5x.) (S-52+) (£-52_)l^} .
The quantities 5 defined in eq. (4-32), are complex conjugate numbers
when (l-a)(l-b) < 0 ; when (1-aKl-b) > 0 they are real and
d - ~ H + - 5
as given in eq. (3-57). A similar statement (with
a
o
,b -+ d)
holds for ^2+ defined in eq. (4-42). Thus the functions /*+(£) are
real for all £ if are complex conjugates and are complex
conjugates, while if at least one pair is real then /+(£) are real
for £ > £ , where a 9
fmax{^1 + , £2+) if both ^1 + , £?+ are real,
V«2+
if £ is real and as complex, (4-108)
if is real and is complex.
Note that one of the numbers
f+(
1) is indeterminate.From eq. (4-A3) we have an alternative factorisation of F(£, q ) , the functions ^ + (q) being given by eq. (4-36) or, alternatively, by
g+(
n) = (n2-l) 1|(n-l)(ad+bc)+ia+c)(b+d)
± [Cn-n1+) (n-n^) (n-n2+) (n-n2J ] 5j . The quantities q^+ , defined in eq. (4-A2), are either complex conjugate numbers or are both real, with
V £ 1 + 5 1 •
and similarly for q 2+ (also defined in eq. (4-A2)). Thus the functions
complex conjugates, while if at least one pair is real then
g+
(q) are real for q > ria , where q^ is defined in a similar way to £^ Ceq.(4-108) with £ q ).
The curve T has several branches (Tarski 1960), but for our purposes it is sufficient to examine only one, namely the branch for which q
-*■
+°° either as £ i 1 or as £ i 1 and for which£
-*■
+°° either as r| 1 1 or as T] 4 1 . Using eqs. (4-11) and (4-36) we obtain the following properties of /+ (£) and g+ (q) which we need to determine the behaviour of .When
(a+b)(o+d)
> 0 ,/ +(£) ^ (£-1)
^(a+b)(o+d)
for £ 4 1 , so that / (£) -► +00 as £ 4 1 . Also,g M
^ 1 + q 1 (a+2>)(c+<i) for n -+ +°° ,so that g+(q) 4 1 as q -*■ +°° . Similarly, when
(a+c)(b+d) >
0 , /+(£) 4 1 as £->+°° and g+ (n) ^ +°° as q 4 1 .When (
a+b)(c+d
) < 0 , g (q) 4 1 as q ■> +°° , / (£) -* +°° asi +
£ 4 1 , and
/+(1) = 1 + [2
(a+b)(o+d)l ^\Xad-bc)^- (a+b+c+d)^] .
We also note that, in this case, either (a+b
) < 0 , and then[a
I < 1 , Ib
I < 1 and/+ (5 ) =
e/±±
d/ lt- g _ '
(4-109)-a
/ 1-b2-b
/ 1-a
/hS„,) = . (4-110)
- 2 +
-c/T^-d/T^
Similarly, when (a+o)(b+d) < 0 , / (£) 1 1 as £ -*• +°° , ^_(n) ^ +°° as n + 1 » and
<7+ (l) = 1 + [2(a+o)(b+d)2 ^\\ad-bo)^-(a+b+o+d)^\ .
We also note that in this case, either (ate) < 0 , and then \a\ < 1 , IoI < 1 and
\ b/ 1 -o'
0 + ( O = -- —
" 1 /T 2 / 2
-av l-o -cv 1 -a
or (b+d) < 0 , in which case \b\ < 1 , \d\ < 1 and
* > 2+) - a-
^
3
,__
-b/ 1 -d2-d/ 1 -b2
When (a+b)(o+d) > 0 and (a+o)(b+d) < 0 , we find that <7/1) > 1 .
For, if (a+o) > 0 , (b+d) < 0 , then \b\ < 1 , \d\ < 1 , and (a+b+o+d)2 - (ad-bo)2
= C(a+i)(l+d) + (ö+d)(i-ib)][(a+ib)(i-d) + (ö+d)(l+2?)] > 0 , and similarly if (a+c) < 0 , (b+d) > 0 . In the same way, if
(a+b)(c+<f) < 0 and (a+o)(b+d) > 0 , then / / l ) > 1 •
When (a+b) (o+d) < 0 and (a+o) (b+d) < 0 , / (1) and g + (l) are either both greater than 1 , both equal to 1 , or both less than 1 .
We are now in a position to sketch the required branch 1/ and the region R above and to the right of it for which q ) > 0 . We do not consider the degenerate cases for which at least one of the quantities (a+b) , (o+d) , (a+o) , (b+d) is ,zero; these can be
obtained by modification of the above analysis. There are then four ca s e s : (la) (a+b)(c+d) > 0 , (lb) (a+b){c+d) > 0 , (2a) {a+b)(o+d) < 0 , (2b) (a+b){o+d) < 0 , {a+c){b+d) > 0 ; ia+c){b+d) < 0 ; (a+c)(b+d) > 0 ; (a+c) (b+d) < 0 .
The branch I* and the region R (subscripted according to the case
under consideration)are sketched in fig. 4-4 for these four cases. In
case (2b) the intersections of with the lines £ = 1 , n = 1
2 2
depend on the sign of (a+b+o+d) - {ad-bo) . Regions R and R ^
are subcases of the region
R1 = {(?, n) I € > l, n > / +(
5
)> , (4-111)while regions R ^a and R^ are subcases of the region
R2 = {(c, n) I
5
a < £ < l, f+(£) < n < /_(£)}u {(£, n) I
5
> 1, n > f +(£)} • (4-112)We now give the arguments used to establish the behaviour of .
For case (la), we note that the functions / + (£) are defined and differentiable for £ > 1 , and /_(£) < / (£) ; a similar statement
holds for the functions ^ + (ri) • Knowing the asymptotic behaviour of
/ + (£) , we deduce that / (£) > 1 for £ > 1 ; for otherwise there would be two real values of £ for which F(£, 1) = 0 , which is
impossible. Next, / (£) cannot take the same value for two different
values of £ . For if f + ( ^) = f+ » with 5^ < 9 then / ^ ( O
(-V1) S
Fig. 4-4.
The branch V ^ of the curve T for which F(£,
n)
= 0 , and the corresponding region R for which F(£, n ) > 0 , in the four casesthen g + [f+ {Zm }} = g _ [ f + i^rn)) » which is impossible. Thus / + (£) is
monotonic decreasing on (1, +°°) , and similarly ^ + (n) is monotonic
decreasing on (1, +°°) and they are inverse functions. The
arguments used to establish the behaviour of for the other cases
are similar but slightly more complicated. For case (lb), we use the
additional facts that ^ + (1) > 1 , 0'+ (rla) = 9_ (n ) together with the
asymptotic behaviour of (ri) and the behaviour of g_(.r\) as r\ i 1
Further, we use the fact that (/+ (£)-n a) - 0 for £ > 1 , which
follows by analogy with eq. (4-A7). The arguments leading to the
behaviour of for case (2a) are similar to those for case (lb).
For case (2b) we use the additional facts that (ö'+ (n)-C ) - 0 for r) > 1 and (i7+ (ri)-£ ) - 0 for r)a - >1 < 1 . These relations follow
from eqs. (4-A7) and (4-A12). Further, we use the relations
(/+ (£)~na) - o
for£ > 1
and(f± (?)-na)
-0
for5 £ < 1
whichI
follow by analogy with eqs. (4-A7) and (4-A12). In this case the exact
ordering of the points 1 , <7+ (l) and g + (ri ) and also of 1 ,
/ + (1) and f + (na) varies depending on the values of a, b, a and d
Our object now is to express I(x, y) as a double integral over
the region R or R ^ or, when this is not possible, as a double
integral over the region R ^ or R plus a single integral. To do
this for the case when (a + b ) ( o + d) < 0 requires some additional results. When (o+d) < 0 , (a + b) > 0 and, in addition, a > 1 , b - 1 , we have
£2+ = -1 + j[(l+ö)2(l-<i)2 + (l+d)2(l-c)?]2 > -1 ,
?1+ = - 1 - |[(l+a)^(i-l)^-(l+i)2(a-l)tl2 £ -1 ,
and thus £ > * Hence, if (a+d) < 0 and max{a, b} > 1 , it follows from eq. (4-108) that = £2+ . If (a+d) < 0 and
(a+b) > 0 , but max{a, b) < 1 , it is convenient to define angles a , 3 , Y , (S by
a = arc cos a , 3 = arc cos b , Y = arc cos a , 6 = arc cos <2 , (4-113) where
0 <
a
< tt ,o
<3
< tt , o < y < t t s 0 < 6 < tt .(These angles are similar to those used by Karplus, Sommerfield and Wichmann 1959.) Then
£1+ = -cos(a±3) ,
^2± -c o s(y±6) , n 1+ = -cos(a±Y) , ^2± = -cos(3±(S) .
The condition (o+d) < 0 is equivalent to tt < (y+<5) < 2tt , while (a+b) > 0 is equivalent to 0 < (a+3) < tt . If, in addition,
(a+3+Y+6) 5 2tt , then 0 < (y+6-tt) < tt - (a+3) < tt , and so
c o s(y+<5) - cos (a+3) and 5 • If » however, (a+3+Y+($) > 2tt , then < £1+ • particular, when a + 3 + Y + ö = 2TT, then
^1+ ~ ^2+ and ^±^1+) " -f±^2+^ = ni+ = n2+ *
We have now shown that, when (a+d) < 0 and (a+b) > 0 , ^a = ^2+ ^ max(a, b} > 1 or if -1 < a - 1 , -1 < b - 1 and
(a+3+Y+ß) - 2tt . However, = £ if -1 < a 5 1 , -1 < b < 1 and (a+3+Y+ö) > 2tt . Also, when (a+d) < 0 , (a+b) > 0 and y < -1 ,
/_(£)
______b _ ___ ^ ^ (4-114)
for < £ < 1 • To prove eq. (4-114), note that /+(£) > -1 for
K
-K
< 1 , so that, from eq. (4-12), F(E,, y
) < 0 forE,
5E,
< 1oc cx
and
y
< -1 . That /+ (£) > -1 for 5E,
< 1 may be seen as follows.The argument above shows that £ > -1 . Similarly for case (2b),
ria > -1 , which shows that for this case /+(£) > -1 for E,^ 5 E, < 1 .
For case (2a) this result follows directly from the properties of
f + (E,)
summarized in fig. 4-4.We are now in a position to give the explicit form of
I(x, y)
for the various mass configurations.
I: (