MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 549 



02 is there unit, the curve must there be inclined at 50 c (45°) to the line of 

 abscissa?. Also, since the values of 02, with their signs changed, are those of 

 (— t\ it follows that any straight line drawn through to cut the curve on 

 the one side, cuts it also at a like distance on the other side of the origin. Since 

 the curve is nearly straight at 0, the second derivative of 02 being there zero, 

 and since is nearly straight, and horizontal at A, the curves and must 

 meet each other on some ordinate not far from the value t = 1. Now the values 

 of the four functions when t = 1 are, 



01 



= 



1-04169 14703 



01 



= 



1-00833 60892 



01 



= 



0-50138 91645 



01 



= 



0-16686 51044 , 



so that the difference 01 — 01 is only -03335 53811 ; dividing this by its first 

 derivative, viz., 01 — 01, we find the correction -03, and thus 1-03 is nearly the 

 value of the abscissa of the intersection of the two curves ; the true value of this 

 abscissa obtained by the subjoined computation is P = 1-03841 56373, and those 

 of the corresponding functions are, 



0P = 1-04848 12871 



P = 1-04848 12871 



P = 0-54089 53047 



P = 0-18688 01785 . 



0* 



0' 



0* 



H* 



t 



1-04169 14703 



500 59531 



22 56251 



45375 



352 







1-00833 60892 



3125 07441 



7 50893 



22563 



340 



2 



0-50138 91645 



3025 00827 



46 87612 



7509 



169 



2 



0-16686 51044 



1504 16749 



45 37512 



46876 



56 



1 



1- 

 •03 



104692 76213 



153 18679 



1 87728 



1027 



2 « 



1-03966 42131 



879 41920 



64338 



526 



2 



0-53210 87763 



873 31794 



3 69356 



180 



1 



018236 52239 



446 97137 



3 66794 



1034 







1-03 



■0084 



1-04847 83649 



29152 



1 



1-04846 48918 



1 63563 







0-54087 89095 

 1 63561 



1 



0-18687 17205 



84377 



1 



1-0384 

 •00001 56 



1-04848 12802 

 70 



1-04848 12481 

 391 



0-54089 52656 

 391 



0-18688 01583 

 202 



1-03841 56 



00373 



1-04848 12872 



1-04848 12872 



0-54089 53047 



0-18688 01785 



1-03841 56373 



