210 Dr. A. A. Bobb : Graphical Solution of Differential 



Further, since PN n = R, l A, i it is evident that the relation (4j 

 is satisfied and so the A curve is the curve we have been 

 seeking. 



On the other hand, the N curve is one whose polar equation 

 with P as pole is a solution of the differential equation (2), 

 and is more convenient in practice than the A curve. 



The giving of! different values to* the constant of inte- 

 gration C has the effect of shifting the origin of coordinates 

 on the squared paper along the axis of X,and, since this has 

 no influence on the process of solution, we may take any 

 value of G we like. 



The position of the point P with respect to the curve DE 

 is important, however, since different positions correspond 

 to different solutions of equation (2). 



If we desire to obtain a solution of the differential 

 equation so as to satisfy given initial conditions, we may 

 suppose that we have initially 



P=Po 



^P = (*p\ m 



Now in the usual method of deducing equation (3) given 

 in books on the differential calculus, it is shown that the 

 intercept between the point of contact A and the foot N of 

 the perpendicular from P on the tangent at A is equal to 



the value of -j- at this point. 



But this intercept is equal to R P and so we have only to 

 take an ordinate A R equal to p Q on the curve on the 



squared paper and measure off a length R P equal to j ~ ) 



\ aft)/ o 



along a line parallel to the axis of #, in order to adjust the 

 squared paper over the dot so as to give the required 

 solution. 



Part II. 



By a slight modification of the method already described, 

 it is possible to obtain a solution of the more general tvpe of 

 differential equation 



r/ 2 V /7V 



^f/rv^+rfv,-*'). . . . 0) 



Equations of this form also occur in problems connected 

 with wireless telegraphy and, in practically important cases, 

 we generally, though not always, have 



(£(£) = a sin ml. 



