

476 MR. S. 8. HOUGH ON THE OSCILLATIONS OF A 
ee (Ce Jue BEE C (Cl Biv) 
2 = ow./1 — 
o*| a —(C— A) —B) } 
/ See VB) = ABH Oa Bae) 
2 — (CA) (Ci) ia 
—— oS (Ei) as es) (lg) a? (ll > 2K) saves 1) ey i een) 
therefore 
A= +o0(1 + E). 
(CAC) 
AB 
terms, an approximate value of the second root, correct to first powers of €, €, is 
given by 
Again, dividing by \? — ? in (8), and putting \? = #* in the small 
gc(C — B) — gA 
(C — A)(C — B) 
AB 
«(C= A= B) 
2 9 AB 
ABW — (C — A)(C — B) w& = — ea? 

—1 

(C= a) = pe ae) 




— €0~ (© — A)(C —B) a ? 
AB 
or 
a SC Say ces G = Craw 
Oak Cap 
J pe pen aa ee 8 (a) 
to the same order of approximation. 
This approximation involves the assumption that ¢,, €, are small compared with 
GC =A Cr-=B. 

; the approximate value of the root will, however, be the same if we 

iNew ts B 
C—A C—B 
suppose —~— , —=,— to be small quantities of the same order as «,, &. 
pp A B q i 
Let us put 
OESROS, CE 
Ara Se aS aon? 
Retaining only finite terms in (5), we obtain as a first approximation to the roots 
? = w and dh? = 0; also, the independent term in (5) is a small quantity of the 
second order in kj, kg, €, € Thus the root which approximates to \* = 0 will be of 
the second order. Regarding )* as of the second order, and retaining only terms of 
this order in (5), we get 
2, [4ABa2B?] — w?. 4028. {Bey + Ce} (Ax, + gCe} = 0, 
