26 
ME. G. T. WALKER OK BOOMERANGS. 
force and a couple about the centre ; the mean normal pressure per unit area may be 
denoted by where X. is a constant that depends on the proportions of the rect¬ 
angle and p, is a number which has been often assumed to be equal to 2. In his 
“ Experiments in Aerodynamics” (‘Smithsonian Contributions to Knowledge,’ 1891), 
S. P. Langley makes this assumption, but if from Table XIV. the value of p be 
calculated by comparing the soaring velocities of 24 X 6 planes, weighing 250 and 
1,000 grms. at inclinations of 10° (the smallest inclination quoted), it proves to be 
27 ; for square planes of the same weight, inclined at 5°, the index is 2*5. From 
comparison of the cases of inclination of 2° and 5° of Table XIL, the value 3‘3 of p 
may be deduced. In the course of the following analysis it will be seen that progTess 
is attended with extreme difficulty unless p = 3, and inasmuch as the constant X is 
at our disposal, we shall be justified in taking p = 3 and choosing X so as to agree 
with Langley’s experiments at the mean value of the velocity under discussion. 
Any error introduced liy an incorrect value of p will be quantitative rather than 
qualitative. 
In addition to the uniform jiressure acting on the recta,ngular plate, there will be a 
couple whose amount may be taken as KV''ci per unit area, k being a constant depending 
on the dimensions of the plate.* The assumption of a velocity potential would lead 
to the value v = 2,t while z/ = 3 is suggested by the previous assumption. In order 
to simplify subsequent proceedings we shall choose the smaller value and deduce the 
value of K from Langley’s experiments. 
We have now to consider the effect of the air on the slightly distorted thin surface 
which represents the boomerang, and in order to surmount the difficulties introduced 
by the fact that the velocities at different points vary, as well as the directions of 
the normals to the surface, we are driven to make some hypothesis. 
Now the effect of the air-pressure upon a plane surface in uniform motion may be 
obtained by integrating over it, provided that we regard the effect due to any small 
portion as proportional to the area of that portion. 
We therefore assume, as a first approximation, that the contribution from any 
element of the distorted surface is the same as if the rest of the surface were in the 
same plane as the element and had the same velocity ; that this assumption, in the 
case of simple distortions, leads to results of the right character, is easily verified. 
The determination of k depends on the fact that if the width of an arm measured 
in the direction of the velocity of the point in question be c, and if J stand foi the 
* See Thomson and Tait, ‘ Natural Philosopliy,’ § 325. The existence of this couple is often stated 
in the form that the resultant thrust on the plate does not act at the centre of figure. Langley finds 
(chap, viii., pp. 89-93), that in the case of a scpaare ijlate the point of application of the resultant 
pressure, when a does not exceed ten degrees, is at a distance from the centre of figure equal to about 
one-sixth of the length of the side, lie quotes JoESSEL and Kummer as having obtained a fifth and a 
sixth respectively as the value of this ratio. 
t Lamb's ‘Hydrodynamics,’ ]). 185 (3) ; Basset’s ‘ Hydrodynamics,’ vol. 1, § 190. 
