I. DIFFERENTIAL EQUATIONS. 561 



the motion of a point describing any line in space with any velocity. 

 Now there are a four -fold infinity of lines in space, and a single 

 infinity of velocities. We therefore see the very general nature of the 

 information contained in the differential equations. So in the example 

 of 13 the statement that all the planets experience an acceleration 

 toward the sun which is proportional to the inverse square of the 

 distance expresses a very general and simple truth, in the form of a set 

 of differential equations, while the integral states that the planets describe 

 some conic section in some plane through the sun, in some periodic 

 time, all the particulars of which statement are arbitrary. 



The characteristic property of the differential equations of mechanics, 

 for the phenomena furnished us by Nature, is apparently that they are 

 of the second order. This , although leaving possibilities of great generality, 

 suffices to limit natural phenomena to a certain class, in contrast to 

 what would be conceivable. For the consequences of the removal of 

 this limitation, the student is referred to the very interesting work by 

 Konigsberger, Die Principien der Mechanic. 



In order to determine the particular values of the arbitrary constants 

 applicable to any particular problem, some data must be given in addition 

 to the differential equations. It is customary to furnish these by stating 

 for a particular instant of time, the values of the coordinates of each 

 point of the system, and of their first time -derivatives, which amounts 

 to specifying for each point its position and its vector velocity for the 

 particular instant in question. This furnishes six data for each independent 

 point, which is just sufficient to determine the constants. Thus if we 

 are dealing with a system of n points free to move in any manner, 

 under the action of any forces, the statement of the problem will consist 

 in the giving of the differential equations 



dt ' dt ' dt*' d 



together with the so - called initial conditions , that for t = t Q , 

 % = V, 2/i = */i - z n = %n 



dx i _ iy no fyi _ r.J no . . ^ _ rJ 10 

 ~dt ~ I* 1 - 1 >~di ~ Ll/lJ ' ~dt " L * wJ ' 



From these it is required to find the integrals 



% = /i(0 2/i = /i(0 *i = /s(0 * = fi-W- 



Cases involving the motion of points whose freedom of motion is limited 

 are dealt with in subsequent chapters. 



WEBSTER, Dynamics. 36 



