184 On Differential Equations 



or, multiplying (1 6) into 4 and reducing 



1 8. This equation is of the second order. It remains to 

 be shown that its solution depends upon that of a linear 

 equation of the same order. To this end divide it b}' p^ and 

 it becomes 



2 ^_ 1 (dpy _^^ _ ,jg, 



which is equivalent to 



^^\l..dp\ + (l.p]\^r = .(19) 



dx \ p dx ) Kp dxj ^ ^ 



^ 9. Now if we put 



1-^ = .. - ^ - .(20) 



p dx ^ ^ 



then (19) becomes 



2a^—+a^V^ + 4iT=0 - - -(21) 



(Jj X 



which if a =: 2 reduces to 



. p^ + v^ + r = - - - (22) 

 di X 



and so to 



or (making 

 to 



fh X J 



d^ W 



dx 



+ r w = - - - - (23) 



20. Now, retracing our st^ps, we see that (1) and (23) are 

 identical in form, and also that (compare article 9) 



