.730 Mr. STOKESS DISCUSSION OF A DIFFERENTIAL EQUATION, &c. 



Neglecting the third term in equation (59), and putting for u its value, we get 



wh 



ere 



—4 + 9V = 2g'-'5'GT - 2.r3 + x% (60) 



ax 



„ _ 63 Mg 1008 M^ 



' " iXM'VS " 31 M~ ^'''^ 



The linear equation (60) is easily integrated. Integrjiting, and determining the arbitrary con- 

 stants by the conditions that |U = 0, and — = 0, when x = 0, we get 



„f . , 12.r' / 12\ I sinoa?N 24 1 



^='iShxr - ■ici? ~ + ( 1 + ^) (•'' ^—\ + -^(1 - cos9,r)l;... (62) 



and we have for the equation to the trajectory 



y = 5m (a; - 2*' + a^') = 5/x (^ + X% (63) 



where as before X =■ x (\ — ,n). 



When F= 0, g = jo , and we get from (62), (63), for the approximate equation to the equi- 

 librium trajectory, 



y= 105'(^+ X'-Y; (64) 



whereas the true equation is 



y = \Q,SX' (65) 



Since the forms of these equations are very different, it will be proper to verify the assertion 

 that (64) is in fact an approximation to (65). Since the curves represented by these equations are 

 both symmetrical with respect to the centre of the bridge, it will be sufficient to consider values of 

 X from to i, to which correspond values of X ranging from to i. Denoting the error of the 

 formula (64), that is the excess of the y in (64) over the y in (65), by S^, we have 



1= - 6X^ + 20^3 + WX\ 



^° , ■., ,,,^ -^dX 



-—= 4(- 3 + 15^+ 10^-).^-;—. 

 dx dx 



Equating —— to zero, we get .^ = 0, j? = 0, ^ = 0, a maximum; .^=.1787, .r = .233, 



^ = - .067, nearly, a minimum ; and *' = ^, 5 = — .023, nearly, a maximum. Hence the greatest 

 error in the approximate value of the ordinate of the equilibrium trajectory is equal to about the 

 one-fifteenth of S. 



Putting |u = Mo + y"i) 2/ = 2/o + 2/i' where mq, y„ are the values of fi, y foT q = os , we have 



{12 /l 12N 24 1 

 — xCl - x) - — H- — Sin g a; + -7(1 - cosoa?) > (66) 

 T W ffl q J 



y, = 5a;(l - x)\\ + x (\ - x)] y^ .". (67) 



The values of mi and y, may be calculated from these formula for different values of q, and 

 they are then to be added to the values of Moj .%> respectively, which have to be calculated once 

 for all. If instead of the mean deflection « we wish to employ the central deflection Z), we have 

 only got to multiply the second sides of equations (62), {66) by |-|, and those of {63), (67) by 1|-, 

 and to write D for fi. The following table contains the values of the ratios of D and y to S for ten 

 different values of q, as well as for the limiting value 7=00, which belongs to the equilibrium 

 trajectory. 



