the Calculus of Variations. 323 



tuted for y in V ; p + ku' for p ; gr + A m" for q ; r + * w w 

 for r ; &c. ; we have 



W = U + */«** (f .+ -0,'+ f -«"+ &* + **j 

 + <Z^2-u'u" + 4t-w //s + &c.\ + &c 



dpdq dq H ) 



= U + kfdx{ Nm + P«' + Qm" + R w'" + &c. } + 

 I- rdx{&i? + 2Buu> + C«' 3 + 2Da«" + ZEu'u" + 

 Fw //2 + &c.) + &c. for the sake of conciseness. 



Now y*d £ P u f = P k —fdx «F: and therefore, since 

 « vanishes at the limits,y^.r Pa' is the same as —fdxuF', 

 when both are taken beween the limits. Hence if we denote 



S*X, 



the fdxVu' when taken between the limits by J dxVu f , 



SIX, f*x, 



we have J dx Pw' = — J dxuP r . Similarly we shall find 



x » *» 



J dxu"Q —J d xuQ" : and so on *. Hence if we substitute 

 the preceding results, we find 

 W^W,, = l^-U,, + kfdx {N-P+Q"- R w + &c.} 



x /( 



+ &c. the terms following those which are written down 

 being multiplied by # 2 , F, &c. Now since the term multi- 

 plied by k may be altered from positive to negative by only 



* It may be necessary to explain the notation made use of in the follow- 

 ing investigations. The expressions/^ V and fdx {N— P' + Q"— &c„} 

 denote the functions whose differential coefficients taken with respect to 

 x are V and N — P'+ Q" — &c. respectively: fdx is considered in these 

 expressions as a mere symbol. For the sake of conciseness it is desirable 

 to have an expression for such a function as fdxY when a certain value^ 

 as x, or x n , is assigned to the variable x contained in it : the expressions 



/**/ f x „ 



J dxY and J dxV are used for this purpose. An expression equivalent 



toy dxV—J dxV very often occurs, and it is therefore convenient to 



f\X, 



express it by one term, as / dxV. In the use made of these expressions 

 %J x n 



it will be seen that I dx and I dx are considered as mere symbols. 



J ' x„ 



2 T 2 changing 



