This is simply the equation for a straight line passing through the points 

 ( V z k )and( Vl ,z k+1 ). 



Integrating (1) once yields 



' , " Z k (x k +1 - ^ , Vl (X - \ )2 {9i 



f(x) = 2d + 2d + c l (2) 



k k 



where c, is the constant of integration. 



Integrating (1) a second time yields 



\ ( Vi " x) \+i (x " "k' 



f M =iJ ^ + 6d k ^,x,c 2 . (3) 



With the requirement that f(x) match the observed data at the points 

 x, and x, ,, the constants c, and c„ may be evaluated from equation (3). 



Thus, at the point x, , 



and at the point x, ., 



.2 

 Vl d k 

 Vl = f( Vl } = - o - + c l Vl + C 2- (5) 



Solving equations (4) and (5) for the values of c. and c_, and substituting these 

 into equation (3), yields the following formula for the cubic spline in the 

 interval x, < x < x, . , 



