158 SECOND DIFFERENTIAL COEFFICIENT 



181. This result may also be found from Art. 176, by 



d u 

 supposing w = always, and therefore -7-3=0; or indepen- 



dently thus. 



From u = 



. f n (du\ dy . (du\ 



it follows that j- }-f- + -7- =0 .................. (1). 



\dyj dx \dxj 



Denote this result for the sake of shortness by 



v=0. 



fdv\ dy fdv\ 

 Hence U- -/ + U- = .................. (2), 



\dyj dx \dxj 



which result, expressed in terms of u, is 



d V. \ / d*u \dy (d*u\ fdy\* fdu\ d*y _ 

 \dx dy) dx + (dy*J \dx) + \dy) ~dtf 



as ~ is already known, this equation will furnish -= . 



Equation (1) is frequently called the "first derived equa- 

 tion," or " the differential equation of the first order ;" and 

 equation (3) is called " the second derived equation," or the 

 " differential equation of the second order ;" the equation u = 

 being called the " primitive equation." 



182. Should the reader succeed in correctly deducing for 

 himself the important equation (3) of the last Article, he may 

 omit the next two Articles, as it seems unnecessary to direct 

 his attention to difficulties he might have felt, or mistakes he 

 might have made. If however he has failed in his attempts, 

 he may compare his process with the following. 



dij 

 In (1), put p for -j-, so that v stands for 



fdu\ fdu 



fdv\ f d*u \ , fdu\ fdp\ , fd*u 



Hence -y- = 37- KP+j-j+ju 

 \dxj \daadyj f \dyj\dxj \dx 2 



fdv\ _ fd^u\ /du\ (dp\ f <f u \ 

 (dy) ~ W<) P + \dy) (dy) + (dy^dx) ' 



