2 3 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



practical means of obtaining their numerical values. The same things are true 

 in the case of differential equations. 



For simplicity in the geometrical representation, consider a single equation 



dx = f , . 

 where x = a at / = o. Suppose the solution is 



2. x = 0(0, 



Equation (2) defines a curve whose coordinates are x and /. Suppose it is repre- 

 sented by figure 2. The value of the tangent to the curve at every point on it 



is given by equation (i), for there 

 is. corresponding to each point, a 

 pair of values of x and / which gives 



-J-, the value of the tangent, when 



substituted in the right member of 

 equation (i). 



Consider the initial point on the 

 curve, viz. x = a, t = o. The tan- 

 gent at this point is f(a. o). The 



, L curve lies close to the tangent for a 



*i 2 short distance from the initial point. 



Hence an approximate value of x 



at / = /i, ti being small, is the ordinate of the point where the tangent at a 

 intersects the line t = /i, or 



Xi = f(a, o)ti. + 



The tangent at x\, ti is defined by (i), and a new step in the solution can be made 

 in the same way. Obviously the process can be continued as long as x and / 

 have values for which the right member of (i) is defined. And the same process 

 can be applied when there are any number of equations. While the steps of this 

 process can be taken so short that it will give the solution with any desired 

 degree of accuracy, it is not the most convenient process that may be employed. 

 It is the one, however, which makes clearest to the intuitions the nature of the 

 solution. 



10.6 Outline of the Method of Solution. Consider equations 10.50 (i) and their 

 solution (2). The problem is to find functions </> and ^/ having the properties 

 (2). If we integrate the last two equations of 10.50 (3) we shall have 



' >t)dt ' 



i. 



The difficulty arises from the fact that </> and \p are not known in advance and 

 the integrals on the right can not be formed. Since </> and \f/ are the solution 

 values of x and y, we may replace them by the latter in order to preserve the 

 original notation, and we have 



