560 NOTES. 



which we may again use to eliminate constants c from 3), so that 

 instead of l) or 3) we now have the system 



. dx dy dz d*x d*y d* z\ 

 y, M, -, -> , f , = 0, 



~) 



=0, 



These differential equations, since the order of the derivatives of the 

 highest order contained in them is the second, are said to be of the 

 second order. In like manner we may continue, and successively eliminate 

 all the constants c x , C 2 . . ., obtaining differential equations of successively 

 higher orders. Reversing the process, each set of a given order is said 

 to be the integral of the set of order next higher. 



Any of the sets of differential equations represents the functions 

 x, y, #, but with the following distinction. If the equations 1) contain 

 constants, to which different values may be assigned, 



6) F t (x, y, 0, t, CJL, c 2 . . . c^ = 0, F 9 (x, y, z, t, c 1? c 2 . . . c n ) = 0, 

 .F 3 0,2/, *, t, c t , c 2 . . . c B ) = 0, 



for every set of values that may be assigned to the constants, a different 

 set of functions is represented, so that we have an infinity of different 

 functions, the order of the infinity being the number of constants contained 

 in the equations. Now the differential equations obtained by eliminating 

 the arbitrary constants represent all the functions obtained by giving the 

 constants any set of values whatever. Thus the information contained 

 in the differential equations is in a sense more general than that contained 

 in the equations 6), in which we give the constants any particular values. 

 If we reverse the process which we have here followed to form the 

 differential equations, we see that every time that we succeed, by 'inte- 

 gration, in making derivatives of a certain order disappear, we introduce 

 at the same time a number of arbitrary constants equal to the number 

 of derivatives which disappear. Thus the integral equations of a set of 

 differential equations of any order will contain a number of arbitrary 

 constants equal to the order of the differential equations multiplied by 

 the number of dependent variables. As an example consider the very 

 simple case of equations 38), 13. 



38") 



d ~ - ' dt* ~ ' 



Integrating these we obtain 



39) x = c t + d 1 y = c 2 t+d 2 , z = c 3 t + d 3 , 



containing the six arbitrary constants c,, c 2 , c 3 , d 1? d%, c? 3 . The meaning 



of these integral equations is that the point x, y, z describes a straight 



line with a constant velocity. But the differential equations 38) represent 



