The integrals in the author's general expressions for the velocity potential Φ of the air motion about a screw propeller may be evaluated by Lommel integrals when the distribution of circulation are expressed in terms of Bessel functions such as I
ν(νμ) and K
ν(νμ).
The decrease of effective angle of attack α
i is calculated for several such types by means of Nicholson's asymptotic formulas. The asymptotic expression for α
i is then obtained in terms of the infinite series such as
or
which the author will propose to call the "Asymptotic Propeller Functions, " where l denotes the number of blades.
The systematic effeet of the number of blades to αi for the same distribution of mean circulation of the total annular element is shown in Fig. 3. The simple theory based on the assumption of infinite number of blades is observed to estimate the effect of the induction too much for the neighbourhood of the axis and too little for the neighbourhood of the blade tip.
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