710 



Mb. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



3. In order to lead to the required integral of (4), let us first suppose that .r is very small. 

 Then the equation reduces itself to 





(6) 



of which the complete integral is 



(id 



and (7) is the approximate integral of (4) for very small values of x. Now the second of equations 

 (2) requires that A = 0, B = 0,* so that the first term in the second member of equation (7) is 

 the leading term in the required solution of (4). 



4. Assuming in equation (4) y = {w - x')- z, we get 



d- 



da^ 



{{x - xyz] +(iz==l3. 



(8) 



Since (4) gives y = {x - ar'y when fi = os , and (5) gives /3= 05 when F= 0, it follows that x 

 is the ratio of the depression of the body to the equilibrium depression. It appears also from 

 Art. 3, that for the particular integral of (8) which we are seeking, z is ultimately constant when ,r 

 is very small. 



To integrate (8) assume then 



z= A^ + Aix + J.,x- + ... = S^,*"', (9) 



and we get 



2 (i + 2) (i + 1) JiX' - 22 (J + 3){i + 2) A^x'*' + 2 (i + 4) (i + 3) A^x'*' + /32^,y = /3, 

 or 



2 I [(J + 1) (i + 2) + jS] ^; - 2 (j + 1) (i + 2) Ai_, + (J + 1) {i + 2) Ai.,\ x' = /3, ... (lO) 



where it is to be observed that no coefficients A^ with negative suffixes are to be taken. 



Equating to zero the coefficients of the powers 0, 1, 2. ..of a? in (10), we get 



(2 + /3) ^„ = /3, 

 (6 + /3) ^, - 12^0= 0, &c. 

 and generally 



{(i+ l)(J + 2) +/3}^,-2(i + l)(j+2)^i_, + (J + l)(i + 2) Ji.„=0 (11) 



The first of these equations gives for A^ the same value which would have been got from (7)- 

 The general equation (11), which holds good from i = I to i = eo , if we conventionally regard 

 A^i as equal to zero, determines the constants Ai, A.,, A^... one after another by a simple and 

 uniform arithmetical process. It will be rendered more convenient for numerical computation by 

 putting it under the form 



^.= K-.H-A^,,.}{i- (,^,^(f^,^^^ |; m 



' When (3> J , the la.st two terms in (7) take the form jt | Ccos 

 (7logj) + i)sm(9logj)}; and if yi denote this quantity we cannot 



in strictness speak of the limiting value of -^ when a* = 0. If we 



give X a small positive value, which we then suppose to decrease 



indefinitely, -j-^ will fluctuate between the constantly increasing 



limits ±j;-«Vl(JC + 9Z))' + (ii)-?C)M, or ±x->^l\ii(C^ 

 + Z)-)5 , since ysVC/S-j). But the body is supposed to enter 

 the bridge horizontally, that is, in the direction of a tangent, since 

 the bridge is supposed to be horizontal, so that we must clearly 

 have C^ + Z)= = 0, and therefore C = 0, D = 0. When /3 = J the 

 last two terms in (7) take the form j-* ( £ + /■ log .»•), and we must 

 evidently have E = <S, F = 0. 



