538 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



A2T = AT 2 + 2 AT . AT 

 A2T = 2AT 2 + AT 2 

 A2T = AT 2 + 2 AT . AT , 



therefore, on measuring a distance of 2T along the line of abscissae, we reach the 

 point at which the line of At is crossed by that of At . 



Similarly, if we put 2T for t, and T for u in the same equations, we have 



A3T = 2AT 3 + 6AT 2 . AT + AT 3 



A3T = 3AT 3 + 3AT 2 . AT + 3AT . AT 2 

 A3T = 3AT 3 + 3AT 2 . AT + 3AT . AT 2 , 



showing that the functions A^and At have again become equal to each other; 

 wherefore the distance, measured on the line of abscissae, from the intersection 

 of the curves 1 and 2 at the origin to their next intersection, is 3T. 



33. It may be shown that this order of intersection continues indefinitely in 

 the following manner. Let w and x be two roots of the equation At — At = o, 

 then is w + x also a root of the same equation ; 



for A(w + x ) = A w A x + Aw ■ A x + Aw . A^ 

 and A( w + x ) — A w A x + A w • A^ + A^ • A x » 

 but by hypotheses, Aw = Aw, Ax = Ax, wherefore the above values may be 



written 



A(w + x) — A w - A x + A^ • A x + Aw ■ A^ 



A(^ + x ) = A w - A x + A w • A x + A w • A x » 



which are identic. From this it follows that the equation At — At = o is satisfied 

 by every value of t which is a multiple of 3T. 



34. The same curves, viz., those representing the functions A£ and At, inter- 

 sect each other on the opposite side of the origin, and on ordinates situated also 

 at the distance 3T from each other; for, according to what has been shown in 



article 29, 



A( — w) = A^ 2 — A w • A w 



A( - w ) — A?" 2 - A^ • A w i 



wherefore if Aw be equal to Aw, A(—w) must also be equal to A( — w), and 

 thus the two curves intersect each other on all ordinates placed at distances 

 =b 3rcT from the origin. 



35. Again, if w be such a value of t as to make Am = Aw, and if T be added 

 to w, we have A{w + T) = A(w + T). For 



&(w + T) = Aw . AT + Aw . AT + Aw . AT 

 A(w + T) = Aw . AT + Aw . AT + Aw • AT ; 



but, according to hypotheses Aw — Aw, and (article 31) AT = AT, wherefore 



