546 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



values of the quaternary functions are 1, 0, 0, 0. From these the values of the 

 functions HI; 0*1; 



0* 



\7}t 



0* 



0* 



t 



1-00000 00000 

 41667 



0-00000 00000 



•10000 00000 



833 



0-00000 00000 



500 00000 



14 



0-00000 00000 

 16 66667 



o- 



1 



1-00000 41667 



1 66667 



2 50000 

 1 66667 



41667 

 

 



0-10000 00833 



•10000 04167 



8333 



8333 



4167 



833 







0-00500 00014 



1000 00083 



500 00208 



278 



208 



83 



14 



0-00016 66667 



50 00002 



50 00004 



16 66674 



7 



4 



1 



0-1 

 •1 



1-00000 66668 



13 33336 



10 00004 



3 33338 



41670 



1 







0-20000 26666 



•10000 66667 



66667 



33333 



8333 



833 







0-02000 00888 



2000 02667 



500 03333 



2222 



833 



167 



14 



0-00133 33359 



200 00089 



100 00133 



16 66778 



56 



17 



3 



0-2 

 1 



1-00033 75017 



0-3000202499 



0-04500 10124 



0-00450 00435 



0-3 



-1 and • 1 are computed by assuming the addition St — ■ 1 ; in this first com- 

 putation the numerous zeroes are omitted. From these, by making another addi- 

 tion St — • 1, the values of the functions of *2, and thence, again, those of the 

 functions of 3 are computed. 



45. The functions of the sum t + u are given in terms of those of t and u 

 separately, by the four following equations : — 



0(Utt) = Qt.0M + 0<.0«+[3]«.0« + Qi.[T]M , (2) 

 [T|(« + M) = 0e.|T]M + 0*.0tt + [o]< g m + < . w , (3) 



[T]((5 + M) = 0«.0M + 0<.0« + 0i.0M +0^.0m ; (4) 



and similarly those of the differences t — u are 



[7|(i-M) = [o]i.0M-0f.0M + 0«.0tt-0*.0M, (5) 



0(«- m) = 0*. 0u-0i.0« + 0< . 0m-0<. Q« , (6) 

 |T](t_M) = 0«.0tt-0«.0u + 0«.0w-0«.0w , (7) 



0(«-w) = 0<.[o]u-0*.[3]u + 0*.0tt-0*.0M. (8) 



These follow at once from the development of the functions of t + u and 

 t — u, by help of Taylor's Theorem. By combining these eight equations in 



