500 Lord Rayleigh on the Resultant 



<j> n known and consider the inclusion of an additional point,, 

 we see that 



*»+i(°')=i 4>j*)fol* ( 6 ) 



J<r-« 



By means of (6) the various functions may be built up in- 

 order. 



We start from <f>\{a). This is zero, unless < a < a, and 

 then is unity. Hence between and a 



<^ 2 (cr)=l <^) 1 (cr)(Zcr/a = cr/a. 

 Jo 



If cr lies between a and 2a, 



( J a 2 a a 



<l>i(<r)d<rla=~ . 



. <r-a a 



Thus 

 2 (o-) = O, (<r<0); <t> 2 (<r) = (r/a, (0<a<a); 







2 (o-) = O, (C"<U); <t>2{.<r) = a/a, (0<a<a); ~^ 



,0 2 (o-) = (2a-<r)/a, (a<a<2a); 0»(<r) = Q, (2a<o-)J ' 



by which <£ 2 is completely determined; and it will* be seen 

 that there is no breach of continuity in the values of c/> 2 

 itself at the critical places. These values are symmetrical 

 on the two sides of <r = a, and can be represented on a 

 diagram by two straight lines passing through a = and 

 a = 2a, and meeting at a = a. (See fig. 1.) 



In like manner we can deduce <£ 3 from </> 2 . If o < 0* 

 <£ 3 =0, and indeed generally <£ n =0. If 0<a<a, 



r i °a da a 2 



U<T) = Jo ~^ = 2a} 



If a < a < 2a, 



4> s (*)= I" °^--f C 2a ^d* = (-Za 2 + 3aa-<r*)laK. 



J<r-a a J a a 



From the symmetry it follows that when 2a < a < 3a, 



<^ 3 ( -) = (3a-(7) 2 /2a 2 . 



When a > \\a, </> 3 (cr) = 0. 



It may be remarked that in this case not only is <£ 3 con- 

 tinuous, but also the first derivative </> 3 '. The representative 

 curves for all three portions are parabolic. The maximum 

 of (f> 3 , occurring at a = Sa/2 } is 3/4. 



