512 Prof. W. B. Morton on the Forms of 



the orbit is increased we should get something corresponding 

 to the parabolic and hyperbolic paths o£ the old theory. Of 

 greater interest is the modification which is reached by 

 decreasing 'the velocity, a direction of change which seems to 

 do less violence to the possibilities o£ the universe as it is. 

 This is found to lead to orbits of a new type which pass 

 through the centre and to which, therefore, the name of 

 4 ' captured orbits " may be applied. In the present note the 

 purely mathematical aspect of the matter is alone considered. 

 The co-ordinates r and 6 are taken in their ordinary mean- 

 ings and curves are drawn to exhibit the relations between 

 them. I leave on one side the question as to the possible 

 bearing of any of the results on observable phenomena in a 

 non-enclidean world where any function of r may be used 

 instead of r. 



The first integral of the equation of motion is 



/ du\ 2 _ „ „ 2 m 



I -mi =2mu 6 — ir+ -yy w + const. 



Here m is the "strength of the centre" or "mass of the 

 sun " in gravitational units and h is twice the area described 

 in unit " proper time " of the planet. It is usual to adopt a 

 system of units in which the constant of gravitation and the 

 velocity of light in space devoid of gravitation are each 

 unity, and the unit of length is the kilometre. The two 

 fundamental unit quantities being regarded as dimensionless 

 constants, the dimensions of mass and length become iden- 

 tical and mass also is measured in kilometres. The mag- 

 nitude of this unit mass is T35 x 10 33 gm. 



The cubic expression on the right-hand side must be 

 positive in the region of motion and the coefficient of v? is 

 positive. It follows that when all three roots are real the 

 actual values of u must lie between the greatest root and + oo , 

 or else between the middle and smallest roots. If only one 

 root is real, u will lie between this and + co , Therefore in 

 every case the range of values will be bounded at one end by 

 a root, i. e. 9 the motion will include at least one apsidal 

 distance. We shall suppose the particle projected from this 

 apse with varying velocities defined by A. Further, if we 

 wish to survey the possible forms of orbits there will be no 

 loss of generality in making the apsidal distance unity. For 

 it is evident from the differential equation that a geometri- 

 cally similar orbit will be obtained provided m and h are 

 increased in the same ratio as the linear dimensions. Since 



