540 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



M- At 



A*- At 



At- At 



t 



+ -53350 71951 



_ 



■65970 01534 



+ 



•12619 29583 



1- 





+ 2523 85917 



+ 



•10670 14390 



— 



•13194 00307 



"2 





- 1319 40031 



+ 



252 38592 



+ 



1067 01439 







+ 71 13429 



— 



87 96002 



+ 



16 82573 







+ 84129 



+ 



3 55671 



— 



4 39800 







- 17592 



+ 



3365 



+ 



14227 







+ 474 



— 



586 



+ 



112 







+ 3 



+ 



14 



— 



17 







+ -54626 98280 







•55131 86090 



+ 



•00504 87810 



1-2 





+ 4 64488 



+ 



502 56824 



— 



507 21312 



•0092 





- 2 33318 



+ 



2137 



+ 



2 31181 



• 





+ 709 



— 



716 



+ 



7 







+ 



+ 



2 



— 



2 







+ -54629 30159 







•54629 27843 







■00000 02315 



1-2092 





+ o 



— 



2315 



-t- 



2315 



- 00 04238 



+ -54629 30159 



— 



•54629 30158 



•ooooo 00000 



1-20919 95' 



'62 



which gives for T the value 1-20919 95762. In the year 1850, being in Constan- 

 tinople, and having leisure, I computed this value to twenty-five places, it is 



T = 1-20919 95761 56145 23372 93856 ; 



its relation to the number 7r = 3 , 14159, etc., might here be pointed out, but I 

 prefer to postpone this consideration until the number t itself arise in the course 

 of our inquiry. 



38. Having now determined all those ordinates on which the curves repre- 

 senting the three ternary functions At, At, At cross each other, we proceed to 

 investigate the distances intercepted by them on those ordinates. In article 29 

 it has been shown that the sum of the squares of these functions exceeds the sum 

 of their products in pairs by e~ l , wherefore 



(At - At) 2 + (A* - A0 2 + (A* - At) 2 = 2e-' 



so that the three lines must keep nearer and nearer to each other as t is taken 

 greater ; and thus they soon merge so closely into one line that it is impossible 

 to delineate them separately. This line into which they merge must evidently 

 be that of which the ordinates are I ■■ e t . 



39. Since for the absciss T we have AT — AT = O, we must, according to 

 the preceding article, have (AT — AT) 2 = e~ T , or 



AT-AT = o T 



and consequently, the intervals intercepted on the ordinates drawn at O, T, 2T, 

 3T, etc., are in continued progression, the common ratio of the progression being 



e -hT = .54629 30158 73601 37743 46503 



