**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**

**/+(?) = (S**2**-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 ■> +°° , / (£) -* +°° as**

**i ** **+**

**£ 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 ,**

**I**

*b*

**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 ,
I*o*I < 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**

*R 1 = {(?, n) I*

**€ > l, n > / +(**

### 5

**)> ,**

**(4-111)**

**while regions ** *R ^a* **and ** *R***^ are subcases of the region**

*R 2 = {(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 cases**

**then ** **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 **

**for**

**5 £ < 1 **

**which**

I

**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**

*-*

**K**

< 1 , so that, from eq. (4-12), **K**

*F*

**(E,, y **

**(E,, y**

*) <*0 for

**E,**

5 **E,**

**E, **

**E,**

*<*1

oc cx

and

**y **

**y**

*<*-1 . That /+ (£) > -1 for 5

**E,**

< 1 may be seen as follows.
**E,**

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.
**f + (E,)**

We are now in a position to give the explicit form of

**I(x, y)**

**I(x, y)**

for the various mass configurations.

I: (

**a + b**

) > 0 , **a + b**

**(a+d) **

**(a+d)**

*>*0 . From eq. (4-94) it follows that

**dE,dr\**

**dE,dr\**

**1**

**rr**

**rr**