234 
MESSRS. R. H. FOWLER AND C. N. H. LOCK ON 
Since y = 0 when t — 0, we may write u = K— XQ£, where K is the complete 
elliptic integral of the first kind to modulus k. Treating Qt as independent variable, 
the final form of our solution contains four constants a, X, k and g, which must be 
completely determinable in terms of s, q and b, so that there must be an independent 
relation between a, X, k and g by which any one can be found when the other three 
are known. To determine them it would be necessary to solve (5), the original cubic 
in y 2 . To analyse the experiments, however, we have to solve the inverse problem 
of determining s, q and b when a, X, k and g are known. In practice g is small so 
that, as a first approximation, we may use the following simplified form of ( 10 ) :— 
y = a cn (K — .(14) 
where a, k and X may be treated as independent. By fitting a curve of this type to 
the curve of observed values of y (sin Jri) against Qt, we can determine the constants 
a, X and k. It is at once clear that a — sin where a is the maximum yaw, but 
we shall continue to call this constant a for shortness. The value of g can then be 
obtained from the identical relation in terms of a, k, X, and the curve re-calculated by 
formula ( 10 ) if g is large enough to make it worth while to do so. After that the 
values of a, X and k could be re-adjusted and the process repeated. Theoretically, 
we could presumably arrive at the precise solution in this manner by a limiting 
process. Practically, in nearly every case, the first approximation with ^7 = 0 is all 
that is required. 
The values of s, q and b are given by simple formulae in terms of a, X, k and g. From 
( 8 ) and (9) we get 
l/s— 1 = 4 q (o/Jrf ‘ + a?f 2 — a 2 h 2 — h) ; 
on substituting for q, h 2 and f 2 from (11)—(13), this reduces to 
l/s — 1 = 4X” ■ — cos 2 k + 
cos 2 k (d 2 —3g~) [ 
2 I 5 
1 ~9 J 
• • (!5) 
where k — sin «-.* In practice either a or cos k, or both, are small and g is of the same 
order as a ; the second term inside the bracket may then be neglected in determining s, 
in which case the value of g is not required. This, as we shall see, is really a consequence 
of the smallness of b, its mean value in practice being about 0-015. For q write 
equation (7) in the form 
4 ( Z+ = 4 q(Jrf--Jr+f-), 
l—a 
and substitute for q, h 2 and f ' 2 in the right-hand side, gettin 
1 
4X 
1 — a? ' a 2 l 
4g — — --- + —— -j sm" k + 
cos 2 k (a 2 — 2f/) \ 
'1 —g 2 J 
16 
* This k will not be confused with the k of (A), loc. cit., p. 328, which is the damping coefficient depending 
on the cross-wind force. 
