. \MTI.I:. 

 wh r= ~. 



</a 



I function uf tin- two in: variables 



/>. and in order that u may have a maximum r minimum 



value 6 and <j> must be taken so as to satisfy ja*p and 



- = 0. It will l>r found on trial that 6 and 6 = 

 </<p 4 4 



values. But it will be found that with these values 

 for and < we g 



r <7V 



"-' ' 



. / d*u \ f c?*?* c7 



;iat -777- js -rri 1S positive and w is 

 \cWc/<j)J dd 2 d$* 



maximum nor a minimum when # = and <j> = . All the 



foregoing is a simple example of the Differential Calculus; 

 we proceed to apply it to tin- M.rhaniral Problem in qu< 



Let &u denote the change in u consequent upon ehai 

 the value of 6 from j to T -f ^ and the value of < tn-m 



- to j + 5<^; then it follows from the preceding 5: 

 tions that 



&u = - 1(8(9)' + 48^8^ + (8<)'J + &c., 



ler the &c. are ineluded 

 ..rd.-r than the second. Now although u is m-ither 



;m nor a minimum when 6 and < arc each ' . 



jiiilibrium thru Ix-rause u i- 



of the first order in 8^ and 8<. (See Art. 261.) Mm. 



as u is neither a maximum nor a minimum the equilibrium 



t be stated to be either stable or unstable ?/;/'///-. *<///>/, 



it is in fact stable with respect to some disphu .- -ments and 



