The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,

or, in an alternative notation,

Let ν(M) denote the number of integers N in the range (0 ≤ N ≤ M) for which

where {y} denotes the fractional part of y. It is a convenient but trivial restriction to assume always that θ and φ lie between 0 and 1. There is a well-known result, which will be called ‘Weyl's Theorem’, that if θ is irrational,

The use of Consol at long ranges in ships is hampered by the lack of facilities for accurate plotting. Of necessity ocean charts (scale 1:1,000,000 to 1:12,000,000) are used; and on these it is not practicable to plot running fixes, lay off position lines from celestial observations or to compare a succession of cross bearings with any degree of accuracy.

When Consol bearings can be plotted on a large scale chart and used in conjunction with other position lines, such as Marcq. St. Hilaire, lines of soundings, m.f./d.f., &c, they are of assistance in defining an observed position; this is especially so in conditions of poor visibility, such as prevail in the approaches to the British Isles and in the North Atlantic.

The present-day trawler carries more aids than any other vessel of comparable length and tonnage and, more often than not, more than the majority of big liners and cargo ships. She carries a powerful wireless transmitter giving an output in the neighbourhood of 100 watts in the modern sets; a main receiver with a standby, or emergency one, for listening in to ships of the same Company for information regarding the state of fishing in their areas; and one or two direction finders, one of which (known as a ‘fish snatcher’) is tuned to the trawler wave band to take snap bearings of rival ships who may be on a better living and foolish enough to say so. So far as Consol is concerned the direction finder is used to ascertain the sector in which the vessel is, should the accuracy of the dead reckoning position be in doubt.

In many technical problems on conduction of heat involving convection, radiation, or evaporation at the surface of a body, the flux of heat at the surface is known empirically as a function of the surface temperature with reasonable accuracy. The thermal properties of the body also vary with the temperature, but in many cases the nature of this variation is completely unknown, and in others it is slight over the range of temperature involved. Thus it seems worth while studying problems on conduction of heat in a medium with constant thermal properties and with heat transfer at its surface a given function of the surface temperature. Mathematically such problems occupy an interesting position between the classical linear theory and the general case in which both the differential equation and the boundary conditions are non-linear.

The introduction into both the Royal Navy and the Royal Air Force of the air navigational equipment that is at present envisaged emphasizes the need for a reliable and accurate compass. Further, since no compass can be expected to give completely accurate information in all conditions of use, the errors that are likely to arise and their magnitudes must be known. Before an error can be determined, it is first necessary to define what may be regarded as an error, and then to determine its magnitude as accurately as possible. Gone are the days when an expression such as ‘plus or minus two degrees’ would be regarded as satisfactory. The determination of compass accuracy to a tenth of a degree is now an accepted requirement.

It might reasonably be considered that any discussion of interplanetary navigation at the present moment is slightly premature. So of course it is, from the practical point of view, since no well-informed person seriously imagines that space-travel will be possible for at least twenty or thirty years, despite the colossal efforts which are now being devoted (unfortunately for quite other purposes) to the solution of its engineering problems. Nevertheless the subject is one of peculiar fascination—which is a completely sufficient excuse for discussing it—and the navigation of guided missiles into astronomical space, which will precede the manned exploration of the planets, has of course already begun and will continue on an ever-increasing scale during the next decades.

The need for the rapid reduction of astro-sights has been constantly stressed for use in high speed aircraft outside radio coverage. The method described here is a combination of precomputation of altitudes and graphical representation of intercept from a given position. By this method a fix can be plotted at the instant of time for which it has been precomputed.