718 



Mh. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



14. Let us first examine the progress of the numbers. For the first two values of /3, z 

 increases from a small positive quantity up to oo as j; increases from to 1. As far as the 

 table goes, « is decidedly greater for the second of the two values of /3 than for the first. It 

 is easily proved however that before oj attains the value 1, z becomes greater for the first value 

 of /3 than for the second. For if we suppose x very little less than 1, f(\ - x) will be ex- 

 tremely small compared with <p{x), or, in case (p^oc) contain a sine, compared with the coefli- 

 cient of the sine. Writing Wi for I - x, and retaining only the most important term in /(*), 

 we get from (21), (25), (26), and (28) 



/(x) = 



^\ 



4r cosrTr 



-a^r 



■- — x} log - 

 32 



/3' 



a?i 



p(e 



+ 6" 



-.r,Jsin(^^logi-j (37) 



according as /3 < |:, /3 = ^, or /3 > :j; ; and ;:; will be obtained by dividing f{x) by V nearly. 

 Hence if ^ > /32>/3i >0, sr is ultimately incomparably greater when ji = (ii than when fi = (ii, 

 and when /3 = ^, than when /3 = 1 Since /(O) =J„ = j3(2 + ^)-' = (2/3"' + 1) "', /(O) increases 

 with /3, so that f{x) is at first larger when /3 = ^2 than when /3 = /3„ and afterwards smaller. 



When /3 > ^, sr vanishes for a certain value of x, after which it becomes negative, then 

 vanishes again and becomes positive, and so on an infinite number of times. The same will be 

 true of T. If p be small, f(x) will not greatly differ, except when x is nearly equal to 1, 

 from what it would be if |0 were equal to zero, and therefore /(,?;) will not vanish till x is nearly 

 equal to 1. On the other hand, if p be extremely large, which corresponds to a very slow velocity, 

 z will be sensibly equal to 1 except when x is nearly equal to 1, so that in this case also f{x) 

 will not vanish till x is nearly equal to 1. The table shews that when /3 = i,/(.i') first vanishes 

 between x = .9$ and x = 1, and when fi = ^ between x = . 94 and x = .96. The first value of x 

 for which /(■») vanishes is probably never much less than 1, because as /3 increases from ^ the 

 denominator ,u(e'''' + e'f^) in the expression for (p(x) becomes rapidly large. 



15. Since when (i>^, T vanishes when x = 0, and again for a value of x less than 1, it must 

 be a maximum for some intermediate value. When /3 = ^ the table appears to indicate a maximum 

 beyond x = . 98. When /3 = |, the maximum value of T is about 2.61, and occurs when x = . 86 

 nearly. As /3 increases indefinitely, the first maximum value of T approaches indefinitely to 1, 

 and the corresponding value of x to i. Besides the first maximum, there are an infinite number 

 of alternately negative and positive maxima ; but these do not correspond to the problem, for a 

 reason which will be considered presently. 



16. The following curves represent the trajectory of the body for the four values of /J con- 

 tained in the preceding table. These curves, it must be remembered, correspond to the ideal 

 limiting case in which the inertia of the bridge is infinitely small. 



In this figure the right line JB represents the bridge in its position of equilibrium, and 

 at the same time represents the trajectory of the body in the ideal limiting case in which 

 /3 = or K = CO . AeeeB represents what may be called the equilibrium trajectory, or the 



