194 
MR. ARTHUR SCHUSTER ON THE 
obtained from (37) and we thus find equation (33) for these special values of n and cr 
to become 9 1 . 9 ..’ i /_3 0 is 2 \ , 
^ K \ + 5 T *2 y v 10 K 2 5 k 2 ) — I, 
6^ + y (^ + ¥^) + y (K -^3 0 + ^3 2 ) = fy’, 
fm 2 ° + y’ (2/q° + ~ 7 -«: 3 0 ) + y (- 7 -/c 3 ] - /q 1 ) = — y, 
6/c 2 2 + - 7 -y’K 3 2 + y (|/fi 1 -y/f 3 1 + i f-/c 3 3 ) = |y. 
These and all similar equations in which the right-hand side is equated to zero are 
sufficient to determine the k coefficients, each in terms of a series proceeding by 
powers of y and y. I proceed to show how the successive approximations may be 
obtained. If y and y are both zero, the first of the above equations leads to /q 1 = 
and, as the equation must hold for all values of y, this gives us that portion of /q 1 
which is independent of y and y. The remaining equations tell us that there can 
be no other factor k which has a term not containing y and y. The last three 
equations contain /q 1 and they are the only equations out of the complete series 
which contain this particular factor. Substituting its value as far as it has been 
found and neglecting in the brackets all factors except k 2 , because they must all 
depend on y or y, and therefore introduce quantities of the second order, we obtain 
a set of three equations which determines those coefficients which involve the first 
powers of these quantities. We are thus led to 
k 2 — — iVy j k 2 — ts 7 ’ K - = -hy- 
No other coefficients can contain terms depending 011 first powers. If we now write 
down all equations in which /c 2 °, k 2 \ k 2 occur, we may determine the terms involving 
y 2 , yy and y’ 2 . Thus if the above equation involving /q 1 is reconsidered, we find that 
in view of our knowledge just acquired /q 1 must contain terms in y 2 and y' 2 satisfying 
the equation . 1 , 1 / 8 _ 1 2 _ 0 
- lK l ^ 20y 40/ — u - 
The equations for E' 3 , E 3 3 , E 3 \ give 
12/f 3 1 + y / (-$ Ko 1 + -^Ki) +y ^k±+25k 
12«r 3 2 + y (f/f 2 2 + 10/q 2 ) + y — f/q 1 
12^ + W ^ 3 + y(l^ 2 -|/q 2 4 
12k 3 _1 + y / (—/c 2 _1 + 5 k 4 _1 ) + y (jk 2 ~‘ 2 — f« - 4 _2 + 10k 4 ° 
+ 2 -W) = 0 , 
+ 35k 4 3 ) = 0, 
= 0 , 
-¥*2°) = 0. 
The factors such as /q°, k 2 ~ 2 , K.y 2 , which can only contain powers of y and y higher 
than the first may be left out of account in solving these equations, and we thus find 
all terms which contain y 2 , y' 2 or yy. We may proceed in this manner, gradually 
working by successive approximations from lower to higher powers. The following- 
two tables contain the results, including all powers as far as the third, for the two 
typical atmospheric motions represented by the current functions i/zf and \fj 2 2 . For 
convenience of use the // coefficients now replace the k coefficients with negative 
indices, 
