184 Mr. G. A. Schott on the 
Further, -- is a very small quantity (§ 12). It follows 
0) 
from equations (15) that A — A , C—C , D—D can be 
expanded in ascending even powers of — , beginning with 
the second, while .Scan be expanded in odd powers, beginning 
with the first. Bearing this in mind, we see that on the 
right-hand sides of (14) the coefficient of f is of the order 
I — ! and that of ri of the order — in the first equation, 
that of £ of order _ and those of v, and ? of order zero in 
the remaining equations. Now <f%=—^ iqi) = i]^ tg^=^; 
hence we retain the terms of the first order, that is, of order 
f and /} in the first equation, and those of zero order, that is, 
of order f , rj and £ in the others. 
Thus we write A-~A =%A "(^X, B = B ' 2 ^ , and 
neglect C— C 0) D — D , where the dash denotes differentiation 
with respect to the argument in A and B . By help of (15) 
we easily find 
2/3A "=g", 2B '=g +/ 8W, . . (18) 
where V, W are the series already defined (§§ 4, 15). 
§ 18. Collecting our results together and using (6), (11), 
(12) ... (18), we get the following equations of vibration : — 
-fS = -/3eh cos 6. . . . 
*?S-?T =0 
"We integrate the first equation, substitute from it in the 
second, and eliminate f by means of the third. In the 
result we use the equation of steady motion (2) and obtain 
the equations 
(19) 
W "~V" ft)2 + 7 ?( R - Y + ~) =-i/3ehcose. 
