426 Mr. R, V. Southwell on a Graphical Method 



and so on : it is not easy to explain the method in few words ? 

 but its principles should be easy to grasp, if the reader 

 will construct a diagram for himself by means of the fore- 

 going directions. We have now to show that the curves 

 which would be obtained by drawing continuous lines 

 through the points Oabc .... in fig. 1 and O'a'b'c' .... in 

 fig. 3 will constitute the solution of the differential equations 

 (10) and (12). Considering, for example, the portion al/b of 

 fig. 1, we see that the tangents at the middle points of the 

 arcs ab and be will, if these arcs are small, be very nearly 

 parallel to the chords ab, be : hence, to a first approxima- 



d 2 Y 

 tion, the value of —j-y a ^ the point b will be given by the 



expression x 



I — — J = (slope of be) — (slope of ab) , 



= (slope of ?*0 2 ) — (slope of q0 2 ), 

 by construction ; 



9 r • ii 



9 



b 



= — — 2 , since qr = b'2 r , by construction- 

 o 2 



A similar relation obtains at every section, whence it is 

 evident that we may drop the suffixes, and write 



I d 2 Y , m A n -v 



n^n =0 (lD > 



(since OA in fig. i is the quantity denoted by V) ; and by a 

 similar investigation we see that the ordinates HI of a smooth 

 curve drawn through Q'a'b'e' .... in tig. 3 would satisfy the 

 equation 



^ + ? = (16) 



n das 2 k 



Eliminating m from (15) and (16), we obtain the relation 



M b ^)"?k =0, * ■ * ' (17> 



and comparing this with (6) we can see that the ordinates 

 of fig. 1 (PI. VII.) will satisfy the differential equation of our 



