MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 539 



these values become 



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

 AO + T) = Aw . AT + Aw . AT + Aw . AT 

 which again are identic. 



36. Lastly, the curves At and At intersect on the ordinates corresponding to 

 the abscissae w + 2T, for similarly 



A(w + 2T) = Aw . A2T + A"> • A2T + Aw . A2T 

 A{w + 2T) = Aw . A2T + Aw . A2T + Aw . A2T ; 

 but, according to article 32, A2T = A2T, wherefore 



A(w + 2T) = Aw . A2T + Aw . A2T + Aw . A2T 

 A(w + 2T) = Aw . A2T + Aw . A2T + Aw . A2T 



which also are identic. 



Hence it follows that the intersections of the curves take place as under 



of A and A at (1 ± 3n)T 

 of A and A at (2 =b 3n)T 

 of A and A at ± 3nT 



n being any integer number whatever. 



37. This distance T, or rather its triple 3T, bears a very remarkable analogy 

 to the 7T of the angular calculus ; and it becomes a matter of some interest to 

 determine its value in numbers. In order to this determination we observe that 

 the first derivative of At — At is At — At\ the second derivative At — At; 

 while the third derivation brings us back to the original function At — At. 

 Hence we have the subjoined calculation — 



A*- A* 



A*- A* 



A^- A^ 



t 



o-ooooo 00000 



_ 



1-00000 00000 



+ 



1-00000 00000 



o- 



+ 1-00000 00000 





o-ooooo ooooo 



— 



1-00000 00000 



1- 



- -50000 00000 



+ 



•50000 00000 





•ooooo ooooo 





•ooooo 00000 



— 



•16666 66667 



+ 



•16666 66667 





+ 4166 66667 





0000 ooooo 



— 



4166 66667 





- 833 33333 



+ 



833 33333 





000 ooooo 





00 00000 



— 



138 88889 



+ 



138 88889 





+ 19 84127 





00 00000 



— 



19 84127 





- 2 48016 



+ 



2 48016 





ooooo 





00000 



— 



27557 



+ 



27557 





+ 2756 





0000 



— 



2756 





- 251 



+ 



251 





000 





00 



— 



21 



+ 



21 





+ 2 









— 



2 





VOL. XXIV. PART III. 



7g 



