io6 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



tial quotient itself, , of a given function a, we can construct a sheet for the 

 as 



determination of any function of this differential quotient 

 (a) 



'-'() 



The sheet which allows us to derive this function <p(s) from the given function a (s) 

 will also allow us to solve the corresponding problem of integration, viz, when <p(s) 

 is given to determine the function a (s) which is defined as function of <p by the 

 differential equation (a) . 



In order to construct this auxiliary sheet we solve equation (a) with respect to 

 and obtain -=- = F(<p) or 



ds 



ds 



(b) 



da = F(<p)ds 



As in the preceding case, we consider <p = x as the abscissa and ds = y as the ordinate 

 of a point, and construct the curves F(x)y = .. . 2,-1,0,1,2,. . . to positive or 



negative integer values of da (fig. 85). 

 When a value of da and a value of ds 

 are given, we have a certain curve given 

 on the sheet and a certain ordinate be- 

 longing to this curve. The correspond- 

 ing abscissa then gives the value of the 



function <p = f ( -^ ) . This gives the 

 \dss 



solution of the problem of differentia- 

 tion. If on the other hand tp (s) is given 

 the corresponding ordinate up to a cer- 

 tain curve, da gives the length ds, for 

 which we have a certain integer increase 

 in the value of the required function a. 

 This leads to a method of determining 

 step by step a series of points at which 

 the function 



_flda. 



ds 



Fig. 85. Divided sheet for determination of the field of a 

 function / 







to 



a = a + I F(<pls))ds 



** So 



has given integer values. The procedure is precisely the same as in the preceding case. 

 We shall consider only one simple example. Let /(^~) = - We shall then 

 determine 



that is, we shall determine simply the lengths ds between the points where the func- 

 tion a has the integer values 1,2,3,. . . Corresponding to equation (b) we then get 



ds 



to 



da 



<P 



