The equation considered is of the type

and f(δ) and g(δ) are polynomials in δ.

There are three cases to consider These are:

(I) The roots off f(z) = 0 are distinct and no two differ by an integer.

(II) Some of the roots of f(z) = 0 are co-incident.

(III) There exists an integral difference between some of the roots of f(z) = 0.

A pilot flying solo is in a somewhat difficult position as regards air navigation. Too much in my opinion has been written about how to navigate an aeroplane for the man with an array of instruments and a comfortable chair and chart table to work at, and not enough for the other poor unfortunate. The pilot navigator (always excepting people like Mollison or Lindbergh, Amy Johnson or Jean Batten) is limited, and has to find his way about mostly by map-reading, but if he does his job properly he should at least find out the wind from the weather people, and use it to calculate his course to steer and his ground speed. These will never be quite right, but will be near enough for him to start off; he can checlr how he is going by his map and make the necessary corrections.

Though the familiar proofs of the classical theorems on the multiplication of series are intuitively evident, they present, on reflection, many tiresome details, and by their piece-meal character tend to conceal the underlying unity of the results.

In the following account the multiplication theorems are all presented as simple consequences of a central formula in the theory of limits.

1. Let the rod be of length l, line density d, mass M = ld. If I is its moment of inertia about one end, then from the theory of dimensions

where C is dimensionless, i.e. a constant, independent of M and l.

A linkage is an assembly of coupled bodies or links whose freedom of movement is restricted, after the fixture of one link, by the constraint imposed by their couplings. Frameworks and mechanisms are particular types of linkage. In this article the freedom on a plane or in space of linkages of an unlimited complexity is investigated by a general method whose application can readily be extended to space of four or more dimensions. The results obtained permit the analysis of any given linkage by mere substitution in the appropriate formula; but although the results are primarily intended to assist the analysis of existing linkages, it is shown how they can be used to a limited extent in the synthesis of new linkages to meet given requirements.