MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 545 



Fourth Order. 



44. According to the principles already laid down, the fundamental functions 

 of the Fourth Order are, 



0*= 1 + — - + — + — _ + etc. 



0* = 



t t 5 t 9 



1 + ^5 + - 9 + 



e 



F 

 * 6 



S* - 1.2 ' 



* 10 



+ 77^ 



....12 



....13 



t™ 



m< = 



* 8 



1.2.3 



+ 



* 7 



+ 



t 11 



+ 



t 1& 



....11 ' ....15 



+ etc. , 

 + etc. , 

 4- etc. , 



and it is clear that the values of the first and third of these are the same for — t 

 as for + 1, while those of the second and fourth merely change their signs. Hence 

 a table of these functions for positive values of the primary can be made to serve 

 also for negative values. 



The construction of the table can be readily accomplished thus. Having pre- 

 pared and titled five columns, four for the functions and one for the primary, we 

 place at the heads of these a set of corresponding values ; in order thence to 

 compute the values corresponding to a new state of the primary, we multiply 

 each by the increment 8t of the primary and write the product in the column 

 belonging to the function of next higher title. These results, which form the 

 second line, are now multiplied^by %8t, the products being placed in the adjoining 

 column ; the numbers entered in the third line are multiplied by %8t ; those in 

 the fourth line by ISt, and the work is carried on until the terms become insigni- 

 ficant. The removals, it must be carefully observed, are to be from column 1 

 to column 1, from t to t, from 1 to £, and lastly from 1 back to t. 

 The scheme of the calculation being this — 



0* 



02 . it 



£0* • W 



\\±\t . it* 



etc. 



0* 

 \T\t . it 



10* . it 2 



etc. 



0* 

 \T\t . it 



£0* • « 3 



etc. 



0* 



0* . dt 



i0i . dt 2 



£0* • it s 



etc. 



t 

 + it 



(* + it) 



(<+«*) 



(t + it) 



(t + dt) 



t + it 



An example of the actual calculation is subjoined. Beginning at t = 0, the 



