( 207 ) 
CHAPTER X. 
EQUATIONS OF AN ORDER HIGHER THAN THE FIRST, 
CONTINUED. 
1. We have next to consider certain forms of non-linear 
equations. 
Of the following principle frequent use will be made, viz. 
When either of the primitive variables is wanting, the order of 
the equation may be depressed by assuming as a dependent vari 
able the lowest differential coefficient which presents itself in the 
equation. 
Thus if the equation be of the form 
and we assume 
we have, on substitution, the differential equation of the first 
order, 
(8). 
If, by the integration of this equation, z can be determined 
as a function of x involving an arbitrary constant c, {suppose 
z = <£ (x, c)} we have from (2) 
whence integrating 
(x,c) dx + c. 
If the lowest differential coefficient of y which presents 
itself be of the second order, the order of the equation can be 
depressed by 2, and so on.