IRANSACTlONS OF SECTION A. 525 



descending under gravity with uniform velocity in a straight line inclined at an 

 angle a to the horizon in a resisting medium, the body heing symmetrical about a 

 vertical plane containing the line of descent ; is this uniform rectilinear motion 

 stable or unstable for small displacements in the vertical plane of motion ? 

 Choosing two rectangular axes through the centre of gravity, fixed in the body, 

 the motion is defined by the velocity components u, v of the body, and the angle, 6, 

 which one of the axes makes with the vertical at any instant. The resistances of the 

 medium are reducible to forces X, Y along the axes and a couple, G, about the 

 origin, all of which are functions of u, v, and dOjdf. The equations of motion 

 being written down, the conditions for steady motion are obtained by putting 

 M, V, and 6 constant. If letters with the suffix refer to steady motion the small 

 oscillations or other motions in the neighbourhood of the steady motion are 

 obtained by writing u = Uq + du, v=Vg + dv, 6 = 6^, + 86, where 8)t, dv, 86 are small, 

 and the corresponding expressions for the resistance-components are expressed 

 in the form 



X = X. + SM'^,^ + Sy'^ + S5''^', where 6'=^^^ 

 du dv do dt 



By putting hi, bv, 86 each proportional to e'" and eliminating it is found that the 

 values of n are determined by an equation of the fourth degree, and the character 

 of the small motions in the neighbourhood of the steady motion depends on the 

 nature of the roots of this biquadratic. In order that the steady motion may be 

 stable, the roots of this equation must all have their real part negative, and this 

 leads to five equations of condition which have been given by Routh in his essay 

 on ' Stability of Motion.' Without going into 7jumerical calculations the biqua- 

 dratic character of the equation in « shows that a gliding machine may perform 

 undulatory motions decreasing in amplitude corresponding to roots of the biqua- 

 dratic with their real parts negative, and giving the impression that the machine 

 is longitudinally stable ; but there may be other roots corresponding to unstable 

 motions which may cause the machine unexpectedly to capsize. The conclusion to 

 be inferred from these results is that the future development of the problem of 

 llight lies in the mathematical calculation for different types of machine of the 

 quantities on which longitudinal stability depends, and that until aeronauts have 

 fully worked out the mathematical investigations here sketched in outline it is useless 

 for them to spend their money and risk their lives in further experiments with 

 gliding machines. 



4. On Majj-colouring.'^ By Professor A. C. Dixon, Sc.D. 



The number of ways, N, in which any given map can be coloured with four 

 colours can be determined when this number is known for simpler maps. 



Let N„i, . . . denote the number of ways in which the map can be coloured when 

 simplified by throwing two or more provinces a, b . . . into one. Then the 

 following formulae hold good, and one of them can always be used : — 



When a is triangular and is adjoined by b, c, d, 



When a is quadrilateral and adjoined by b, c, d, e, 



When a is pentagonal and adjoined by b, c, d, e,f, 



N = N„_^ + N,ac + N,,„^ - N„,, _ 

 = four other similar expressions. 



The formulae do not decide the question whether N is always >0. 

 If solid space is divided into compartments in an arbitrary way there is no 

 iiinit to the number of colours needed to distinguish them. 



' The paper will be published in the Messenger of Mathematics. 



