716 Mr, STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION, &c. 



become equal to P it must remain always less than P. Hence v^ admits of but one maximum 

 or minimum value, and this must evidently be a maximum. 



When /3 = - N = 2, and P< h H ... or < 1, and therefore P i(i) has the same sign 



as when /3 is indefinitely small. Hence it is q and not p which becomes a minimum. Equating 



— to zero, employing (27), and putting 2irp = log,^, we find 

 dfi 



^—-^HX + ^'^^-sX)-'- 



The real positive root of this equation will be found by trial to be 36.3 nearly, which gives 

 p = .5717, /3 = J + p° = .5768. If F, be the velocity which gives v, a maximum, u, the maximum 

 value of v^, U the velocity due to the height S, we get 



/7^ c U , 87r/3= S ,^ , 



c 



V, = .4655 ^U, v,= .6288 U. 



12. Conceive a weight W placed at rest on a point of the bridge whose distance from the 

 first extremity is to the whole length as x to 1. Tlie reaction at this extremity produced by fV will 

 be equal to (1 - x)W, and the moment of this reaction about a point of the bridge whose abscissa 

 2ca?, is less than 2cx will be 2c(l - x) a!,W. This moment measures the tendency of the bridge to 

 break at the point considered, and it is evidently greatest when ■», = .■?, in which case it becomes 

 2c{l- x)xW. Now, if the inertia of the bridge be neglected, the pressure R produced by the 

 movino- body will be proportional to (.r? - a")'"'!/, and the tendency to break under the action of a 

 weight equal to R placed at rest on the bridge will be proportional to (1 - .v) x •< (x -x')~~y, or to 

 (x - ,r/')x. Call this tendency T, and let T be so measured that it may be equal to 1 when the moving 

 body is placed at rest on the centre of the bridge. Then T = C (x - ■x^)z, and 1 = C" (1 - 1), whence 



T=i {x - x^) «. 



The tendency to break is actually liable to be somewhat greater than T, in consequence of the 

 state of vibration into which the bridge is thrown, in consequence of which the curvature is alternately 

 greater and less than the statical curvature due to the same pressure applied at the same point. In 

 considering the motion of the body, the vibrations of the bridge were properly neglected, in con- 

 formity with the supposition that the inertia of the bridge is infinitely small compared with that of 

 the body. 



The quantities of which it will be most interesting to calculate the numerical values are z, 

 which expresses the ratio of the depression of the moving body at any point to the statical depression, 

 T, the meaning of which has just been explained, and y, the actual depression. When is has been 

 calculated in the way explained in Art. 10, T will be obtained by multiplying by i{x - x"), and 



then ^ will be got by multiplying 7" by 4 {x - x'). 



o 



13. The following Table gives the values of these three quantities for each of four values of /3, 

 namely ^^, J, J, and f , to which correspond r = 1, r = 0, p = ^, p== 1, respectively. In performing 

 the calculations I have retained five decimal places in calculating the coefficients J„, A^, A,... and S„, 

 Bi, B, ... and four in calculating the series (9) and (30). In calculating (p (x) I have used four- 

 figure logarithms, and I have retained three figures in the result. The calculations have not been 

 re-examined, except occasionally, when an irregularity in the numbers indicated an error. 



