TRANSACTIONS OF SECTION A. 793 



those interested in mathematics, and especially to those engaged in teaching the 

 snbject, a rapid means of describing rolling cnrves as well as envelopes. 



8. The Recent History of the Theory of the Functions tised in Analysis. 



By E. T. WlIITTAKER. 



Some recent advances in the theories of the known analytical functions are 

 described in this paper. The problem of the expression of two variables, con- 

 nected by any algebraic relation, as uniform automorphic functions of a new 

 variable, is discussed ; in this problem, Kleins generalised form of LamtSs equation, 

 from which the functions of harmonic analysis may be derived, is of importance. 



9. The Dynamical Exjjlanation of certain observed Phenomena of 

 Meteor Streams. By Dr. G. Johnstone Stoney, F.R.S. 



The following are the principal results arrived at in this communication : — 



Assiduous observation has brought to light several unexpected events in 

 connection with the many meteor streams which intersect the earth's orbit. Some 

 meteoric streams present difl'erences in the duration and character of the showers of 

 meteors in our atmosphere to which they give rise in successive years. The shower 

 of meteors may be either brief or prolonged over many days ; with some streams the 

 radiant is, roughly speaking, stationary ; in other cases it shifts across a portion of 

 the sky in a variety of ways ; the disposition of the shower about its maximum 

 may be either symmetrical or unsymmetrical ; and so on. Some of these events 

 astronomers have succeeded in reconciling Avith the dynamical conditions under 

 which meteors move ; but there are others which still demand explanation, and 

 the present paper is an inquiry with regard to these. 



The meteors in a meteor stream are bodies of too small mass and too much 

 separated to have any sensible influence on one another's motion. This simplifies 

 the problem, for it justifies our treating each individual meteor as moving in its 

 own orbit round the sun, disturbed only by the attractions of the surrounding 

 planets, and assures us that whatever phenomena may present themselves, they 

 must result from these independent motions of the individual meteors. So long 

 as a meteor is at a distance from all the planets, their influence upon its motions 

 may be investigated by the known methods of dealing with planetary perturbations. 

 It is only when the meteor happens to pass close to some planet, whether the earth 

 or another, that special treatment is required. The study of what then occurs is 

 the aim of the present communication; and it is found that the unexplained 

 phenomena of meteors are either the direct or the indirect outcome of the events 

 which then develop themselves. 



A sufficient investigation is possible by a method indicated by Laplace — a 

 method of approximation which is indeed obvious. Laplace desires us to picture 

 to ourselves a sphere of a certain size surrounding the planet as centre, and 

 accompanying the planet along its orbit. Then a good approximation to what 

 occurs may be obtained by regarding the meteor, while farther from the planet 

 than the boundary of this sphere, as moving in an orbit round the sun under the 

 influence of the sun's attraction only ; and while inside the sphere as moving 

 relatively to the planet under the influence of the planet's attraction only. This 

 is equivalent to supposing that the meteor while in the close neighbourhood of the 

 planet receives from the sun the same acceleration both in amount and direction 

 as he impresses upon the planet. This is evidently an approximation to what 

 actually occurs, and furnishes as good a general result as we can hope to obtain in 

 the absence of definite information with regard to the elements of each separate 

 meteoric orbit. 



With this theorem of Laplace's we shall combine a consideration of which 

 elfective use was made by the late Professor Hubert A. Newton in the inquiry 

 into the origin of periodic comets which is published in the Reports of former 



