Metliod of Integration. 79 



or more briefly in the form 



dx ,hr ■ ■ ■ ■ {^) 



wherein v has been substituted for the expression in 

 brackets in (4). 



Hence equation (i) has been reduced in form to one of 



the first order. Since u is a function of— and ~, it will be 



dx dy 



a function of x and j', therefore we shall have the equation 



dv= rdx + -j-dy. . . • (6) 



Proceeding in the ordinary way, we shall obtain the 

 equation 



v = (p{x + y); . . . . (7) 



or by substitution 



dz dz , , 



dx'-dy-^^'^y)' • • • (8) 



denoting an arbitrary function. From this equation we 

 obtain 



z = (pi{x + y) + ipo,{y-x) . . . (9) 



This method of procedure may also be applied indirectly to 

 the integration of the equation which has such extensive 

 physical application : 



dh jh , ^ 



For X substitute a new variable ^ so that the two are 

 connected by the equation 



s=-a«, . . . • (11) 

 hence, we have 



dz __dz dt, dz dh _ d dz _^d\ 



dx dE, dx dt, ' dx^ dx di, dt,^ 



substituting in (10) we obtain the equation 



dh ^dh 



