356 Mr. A. Stephenson on 
In the limit ?'=sco 
a + (2r-l) a . 
-=* = j- <. 1 numerically, 
; ±(2r-3) * 
so that the series is convergent. The above equations are 
sufficient to determine p and the as. Our special object is 
to find the range of n near //, for which p is imaginary; such 
a value of p indicates a continually increasing amplitude o£ 
<£, and therefore of 0, due to the influence of r. In the 
general case it is not a practicable matter to obtain the 
solution of this problem in finite terms, and we shall there- 
fore assume that a. is small. The coefficients then diminish 
rapidly, and we shall neglect those lying beyond a s and a_3. 
Then to the required degree of approximation 
o 
« ±3 =-"- 8 a ±i; 
and on substituting in (1) and ( — 1) we have 
«i {* 2 -(p-ny- ^n 2 j } = *{4rf-(p+nf}*-* 
a-i{jj?-(p + ny- -^n'j } = j\^n 2 — (p-n) 2 }^, 
whence 
= j { i« 2 - O + «) 2 } { 4» 2 - (p-«) 2 } • 
fju 2 — n 2 being of order a/l this equation is correct to the 
order (a/Z) 3 and determines p to the order (a/l) 2 . Putting 
n 2 =/jb 2 l 1 + kj J and neglecting powers of cc/l above the third 
we find 
2 9_/,2 + ^« 
P — P 72 
57 a 
Thus p is imaginary if k lies between the limits ±3+ -^- r , 
and therefore for a cumulative influence on the motion n 2 
