Showing papers in "Quarterly of Applied Mathematics in 1952"
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), which are Legendre's associated functions of degree n and order two, of the first and second kinds, respectively. It is noted that the same Legendre's associated equation is obtained if we assume instead * = (15) P Since the preceding solution is valid for any integral value of n including zero, a more general solution of ip is expressed by the following four series : P2(cos 4>) p" X , (16) "=0 ,Q2( cost) P"n_1 where A„ are parametric coefficients. Note that P\ vanishes when n = 0 or 1. 1952] TORSION OF A CYLINDER HAVING A SPHERICAL CAVITY 151 Method of solution. Now, consider an infinite circular cylinder of radius a (i.e., r = 1) having a symmetrically-located spherical cavity of radius \a (i.e., p = X), which is under torsion by terminal couples about the axis of the cylinder. For convenience, the centre of the cavity will be placed at the origin. The method of solution is to assume \p as being composed of two parts as follows: i , (17) where \J/0 is the solution of a corresponding cylinder without a cavity, while 4>x is an auxiliary solution which vanishes on the surface of the cylinder. The latter is added to yp0 so that the remaining boundary condition on the surface of the cavity is satisfied by adjusting the parametric coefficients involved in the function. The function \f/0 is io = bar*, (18) where r is a constant. Since \pt vanishes on the surface of the cylinder, it follows that the terminal couple is equal to |x/ira4. The auxiliary function 4*i may be constructed by combining linearly a set of functions each of which satisfies the given differential equation and vanishes on the surface of the cylinder. It appears that the set of functions can be derived from the following particular solution for \p, namely: 2 f* co \p* = ^3Pl(cos
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