498 MR. 8. 8S. HOUGH ON THE OSCILLATIONS OF A 
Let us now change our notation, replacing p’, 
p — b*, p?—c by a’, B, y*; the 
period equation may then be written 
[fA + (B — C—») 0%} (B24 (A — Op) a} — o2(A + B— C)] 
x [8 (8 + )(B + 7) — 40° 
+ ple — y)X [AM + (B— C — ») 0%) 22 (8 + 7) — do 
+0 (6? — 72)? {BY + (A — C—p) 0%} (2 (2 +) — 40%} 
— 2M. (A + B— 0) {n(B—y) +(e — y)} 
+ port (X= 4o2) (2 — ¥*) (8 — 7) =0 
ey 
(44). 
If we put \” = @’, the left-hand member of (44) becomes 
DPIC BC») (A+ B= 
’) 
—#)—(A + B—C)] (e+) (B + 77) — 4088) 
+ OMT) (a eve y 
pane fy? — 8B} 
+ of ( — 7) {A+ B— C —p} fy? — 30} 
— 207 (A+ BC) {u(B =) + (8 = ')} — Spee! (2 =) (B = 7) 
— oF (A+B—-C)| p {(a'-+y’) (P+y’) — 4088+ (2) (yy — 88") —2y"(¥'—B)5 | 
_Fvf(C4+P)\(B+7)—48 B+ (~—B)(P— 302) — 27 (~P—2)5_| 
+ pve! se ey (8? ey) = AaB: iy oye Bil 
fe A IB Vy 302) eB me) ae eon le 
therefore \” = w° is always one root of (44). 
Arranging (44) according to powers of \*, we have 
S| AB (a? -+ y*) (B+ y°) + pA (B? + y’) (2? — y?) + vB (oe? 
Bate UO ry) (Bo ye) =| 
~o rt! (2+y) (@+y) (A+B—C)?—A(A—C—p)—B(B—C—»)} + 4A Bah? | 
=i Cr = Fe eS 5 CS — Op) bayer 
— v(B— y’) {(@ + 7’) (A — C — p) — 4Ba*} 
ap Ay (AS BC) iS? 7) on (a 
_+ 4py (a? — y*)(B’ — ’) a 
+ oth®. (a? + 7°) (8 + vy?) (B—C —») (A—€ — 4) | 
a aes (A BC) A (SC) BBC), | 
= su Be? — y)\(B — © — ») — Avan y) (Aaa 
— of (4026? (B—C —v)(A—C—»)}. 
+7) (B= 7) 
Dividing out by the factor \* — o, we are left with 
