112 



MCEWEN AND MICHAEL. 



02-3 — 01-2 

 X3 + X2 X2 + Xi 



2(182-3 ~ 'S1I-2) 

 X3 - Xi 



is the rate of change in slope, 



whence the slope at any point of abscissa x between (xj, A) and 



(X2, B) is Sl_2 + 



X — 



X2 + Xi 



2(<S2 -3 — <Sl_2) 



X3 — Xi 



X = X2 the slope at (x2, B) is 



S2 — <Sl_2 -\ (iS2_3 — <Sl_2) 



X3 — Xi 



and putting 



(41) 



Similarly, the slope at any point of abscissa x between (X2, B) and 



(X3, C) is S2-3 + 



X — 



(X3 + X2) 



2(<S2_3 — 'Si_2) 



X3 — Xi 



X = X2, the slope at (X2, B) is 



S2 — »S2-3 -\ (82-3 — S1-2) 



X3 — Xi 



and putting 



(42) 



In the same way the slope corresponding to each abscissa between 

 the extremes Xi and x„ is found, where x„ denotes the last group aver- 

 age. In the particular case at hand x is divided into but three groups 

 so that x„ = X3, whence the slopes Si and S3 must be determined by 

 utilizing the rate of change in slope between Si_2 and S2-3 as a means 

 of exterpolating beyond Si_2 and S2-3. Thus 



Xi — X2 

 Oi = Oi_2 -j- 



and 



S3 = S2-3 + 



X3 - Xi 



Similarly equations (45) to (48) are obtained. 



o o , y4 - ys ,p 



(43) 



(44) 



ye - y4 



(45) 

 (46) 



