130 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



Accordingly, these rigorous expressions for the 6 varying elemen^ts, in the present 

 dynamical question, agree with the results obtained by the ordinary methods of inte- 

 gration from the 6 ordinary differential equations (150.) and (151.), and with those 

 obtained by elimination from the equations (113.) (114.) (147.). 



Remarks on the foregoing Example. 



30. The example which has occupied us in the last six numbers is not altogether 

 ideal, but is realised to some extent by the motion of a projectile in a void. For if 

 we consider the earth as a sphere, of radius R, and suppose the accelerating force of 

 gravity to vary inversely as the square of the distance r from its centre, and to be 



= o- at the surface, this force will be represented generally by ^-^- ; and to adapt 



the differential equations (78.) to the motion of a projectile in a void, it will be suffi- 

 cient to make 



U=^R2(-^-^) (173.) 



If we place the origin of rectangular coordinates at the earth's surface, and sup- 

 pose the semiaxis o^ -{• z to be directed vertically upwards, we shall have 



r = ^(R + zY + ^2 + y2^ (174.) 



and 



V = -g.+S^-ii^, (.75.) 



neglecting only those very small terms which have the square of the earth's radius 

 for a divisor : neglecting therefore such terms, the force-function U in this question 

 is of that form (110.) on which all the reasonings of the example have been founded ; 



the small constants jca, v, being the real and imaginary quantities \/ ^i V ~r^' 



respectively. We may therefore apply the results of the recent numbers to the 

 motions of projectiles in a void, by substituting these values for the constants, and 

 altering, where necessary, trigonometrical to exponential functions. But besides the 

 theoretical facility and the little practical importance of researches respecting such 

 projectiles, the results would only be accurate as far as the first negative power (in- 

 clusive) of the earth's radius, because the expression (110.) for the force-function U 

 is only accurate so far ; and therefore the rigorous and approximate investigations of 

 the six preceding numbers, founded on that expression, are offered only as mathe- 

 matical illustrations of a general method, extending to all problems of dynamics, at 

 least to all those to which the law of living forces applies. 



Attracting Systems resumed : Differential Equations of intetmal or Relative Motion ; 



Integration hy the Principal Function. 



31. Returning now from this digression on the motion of a single point, to the 

 more important study of an attracting or repelling system, let us resume the differen- 

 tial equations (A.), which may be thus summed up : 



