712 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



and will be moreover equal to - /3y, if 



m' - m 4- li = (18) 



It appears from (17) that m, n are the two roots of the quadratic (18). We have for the 

 complete integral of (l6) 



y = Ax"' (1 -«)" + B3)°{l - a)"" (19) 



The complete integral of (4) may now be obtained by replacing the constants J, B by functions 

 R, S of ,r, and employing the method of the variation of parameters. Putting for shortness 



a."" (1 - x)" = u, w" (1 - wY = V, 



we get to determine R and .S" the equations 



dR dS 



U-J-+V-- =0, 

 dsc ax 



du dR dv dS 

 da dx dx dx 



du dv , , , 



Since V — - u ■ — = rw — n, we get from the above equations 

 dx dx 



dR (iv dS /3?« 



dx m — n dx m — n 



whence we obtain for a particular integral of (4) 



y = — ^ L" (1 - xY I af (1 - xY dx - a?" (1 - a;)"' / x'" (1 - «■)" dx\ ; (20) 



m -n [ Jo Jo J 



and the complete integral will be got by adding together the second members of equations (ip), 

 (20). Now the second member of equation (20) varies ultimately as x^, when x is very small, 

 and therefore, as shewn in Art. 3, we must have A = 0, B = 0, so that (20) is the integral we want. 



When the roots of the quadratic (18) are real and commensurable, the integrals in (20) satisfy 

 the criterion of integrability, so that the integral of (4) can be expressed in finite terms without 

 the aid of definite integrals. The form of the integral will, however, be complicated, and y may 

 be readily calculated by the method which applies to general values of /3. 



8. Since /J F (x) dx = Jl F (x) dx - j\-' F {l - x) dx, we have from (20) 



y ^ - {x- (1 - xY fl x'il - xY dx- x'il- x)'" fl X'" ( 1 - xf dx\, 



m — n 



+ — ^ \af (1 - xY SI" (1 - i^Y .r" dx - x" (I - x)" fl-' - «)" x'" dx]. 

 m-n 



If we put f{x) for the second member of equation (20), the equation just written is 



equivalent to 



f(.co)=f{\-x) + (p{x), (21) 



where 



fb{x)=-^{x'"{\-xY!l!i:'{\ -xYdx-x-^l -xYPo'V"'{i - xY dx] ...(22) 

 ' m — n 



Now since m + n = 1, 



s^ ds 

 fx" (1 - xY dx = fx («-' - l)"" dx = - /«)-' (w - 1)'" to-' dw = -f . 



