DIFFERENTIAL EQUATIONS OF THE n'" OrDER. 



Differentiating (c) witli respect to .r , we get 



dx dx dx 



dx 



s ^ + t -'^- = 

 dx dx 



e = Ä^- + /3-^-t- 7-7-) and SP' and tp' be eliminated by 



dx 



\dx) \dxl ' \d(f dx dip dx) ' 



(-~?j and |-^| being the differential coefficients of Ç and ;? with respect to 



x^ as far as they involve x both explicitely and implicitely. 



If Q =, Ct ~ ^ 10 ~ -\ 

 dx dx 



means of {b) , this equation will take the form 



F.^ {x , y , p , q , r , s , t, Q , <r , If') ■= . . . . (d); 



and if between (c) , (d) and the given primitive if and >p be eliminated, 

 we shall get 



F3 (Q , '^ , y , z , p , q , r , s , = 

 or, solving with respect to Q and restoring its value 





{e). 



Thus we have the following remarkable result, viz.: that the primitive 

 z = F(if , )/') , (f and being defined as above , gives always rise to a dif- 

 ferential equation, which is linear with respect to its highest partial diffe- 

 rential coefficients of z , whether this diff. equ. be of the second or third 

 order. When the differential equation is of the second order, it will, of 

 course , be the equ. (c) , cleared of (p and tp . 



Another remarkable circumstance, when the diff. equ. is of the third 

 order, is, that we may get more than one diff. equ. from the same primi- 

 tive z = F((f, I/'). This is easy to see, because we might have differentia- 

 ted (c) with respect to 3/ instead of to x , whence we should have obtai- 

 ned in the place of (e) a diff. equ. of the form 



'^ ^ + /3 -^ -\- y -^- =^ f, (x, rj, z, p, q, r, s, t). 



Ex. 6. z = F{<f^ , (f, , ... , (f,) , 



where , as in ex. 3 , every one of the functions f/ contains only one of 

 the definite functions ?t of x and y. We will here examine two cases, 

 viz. 1'"^ when the differential equation is of the (2r — 1)^' and 2'"* when it 

 is of the r"' order. 



