

Development of Arbitrary Functions. 507 



not needed, and indeed it is better to put aside altogether the 

 idea of a curve being wrapped round a cylinder. 



To develop any arbitrary function of x (let us call it y) in 

 normal forms, the real difficulty consists in finding the value 



of an integral such as j y . Q(x) .dx, where Q(#) is some tabu- 

 lated function. If now z is another tabulated function which 

 is the integral of Q(x) the required integral is \y . dz, 



Suppose the values of y for 25 equidistant values of x to be 

 known, from #=0 to x = a (that is, a is divided into 24 equal 

 parts). Let the corresponding values of z be also tabulated, 

 and let a curve be drawn with the values of y as ordinates 

 and the values of z as abscissae ; the area between this curve 

 and the axis of z gives the value of the integral required. 



To illustrate the method we shall show how it is applied 

 in a well-known kind of problem. An arbitrary function 

 y=f(r) of r from r = to r = a is to be developed in Zeroth 

 Bessels ; that is, we must determine the constants A 1? A 2 , 

 &c, in 



f(r) = A 1 J (fj, 1 r)+A 2 J (fjL 2 r) + &c.+A s J {/j, s r) + &c, (1) 



where /Lt 1? jjl 2 , &c, are the successive roots of 



JoM = (2) 



Now it is well known that 



A '= a*[J l( t«) P J/ /W J ° (M ' ^ ' • • (3) 

 or, letting fji s r = x, (3) becomes 



2 1 C^ a 



No w ^x . J {x) .da: = x.J 1 [x) , 



which is easily found for any value of x. 



To evaluate the integral in (4), take the 25 equidistant 

 values of our function from to a and use them as the ordi- 

 nates of points in a curve. The values of the corresponding 

 abscissae are obtained in the following way. The values of 

 fi x a &c. are the successive roots of J (/*a) =0, and are 



fi x a= 2*40483 Take 25 equidistant values of x be- 



ginning with and ending with the value 

 fi s a : the corresponding values of xJ } (x) 

 are the abscissae of points in the curve 

 required to find A s . 



fM 2 a= 5-52008 



fi s a= 8*65373 



/* 4 o= 11*79153 



and so on. 



