2o;.] VARIABLE CONSTRAINTS. Iir 



206. The general integral of equation (23) is 



r = 



If r= r and r v Q when / = o, we have r = ^ + <r 2 , # = <(^i <r 2 ) ; 

 hence 



2 or = (<or + Vo)e<*+(u>ro z> )*""*. (24) 



With #0 = 0, r =i, this equation represents a common catenary.. 

 If V Q = r o>, the equation reduces to the form r = r Q e Mt , whence / = 



The minimum of r in (24) occurs for 



2o) cor -f- V Q 



its value is r^ = VV 2 (# /co) 2 . It is easy to see that such a minimum 

 can occur only when z> is negative and > wr numerically. 



207. To apply the method of indeterminate multipliers to our prob- 

 lem, let the equation of the tube be written in the form 



$(x, y, /) = .# cos to/ j^sina)/ t = o. (25) 



Then we have <f> x = cos o>/, <j>, = sin <o/, </>,= v(x sin <o/4-jy cos <o/) ; 

 hence equation (16) assumes the form 



Bx cos w/ By sin to/ = o ; 

 and the equations of motion (17) are 



mx = A cos CD/, #2)} = A sin to/. (26) 



We have also <f> xx = o, ^ = <$> yx = o, <^> yy = o, ^ x = ^ = w sin w/ r 

 ^ = ^yt = w cos w ^> <#> = > 2 (* cos w/ j^ sin co/) = o. Hence, by 

 ' 



J-5 3r cos to/ j sin co/ 2 our sin to/ 2 toj; cos to/ = o. 

 dr 



Substituting in this equation the values of x,y from (26), we find the 

 linear equation for X which gives 



= 2 to (x sin to/ -f^ cos co/) sec 2 to/. 

 m 



Introducing this value into the equations (26), we have the differen- 

 tial equations of our problem in the form 



x = 2(a(x sin co/ + y cos to/) ^ , y= 2to(^sinco/-f-j ; cosco/) 



COS 2tO/ " COS 2CO/ 



