550 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



This abscissa P, farther than that it marks the intersection of the two curves 

 and 0, is of no immediate interest to us. 



The line having now crossed and passed under the line is next met by 

 the line at a point, see figure 11, of which the abscissa is Oil ; putting II to 

 denote the value of this abscissa, we must have 011 = 011. Now for t = 1'6 

 we have 



1-5 = 1-21157 34084 55475 11794 



01-5 = 1-56338 72208 49435 96389 



1-5 = 1-14083 62067 87772 20785 



1-5 = 0-56589 22342 45381 53295 



hence, by the subjoined process, which is exactly analogous to the preceding, we 

 obtain 



0* 



0* 



0/ 



0* 



t 



1-21157 34085 



3961 24564 



279 50487 



8 93736 



12121 



79 



2 



1-56338 72208 



8481 01386 



138 64360 



6 52178 



15640 



170 



1 



1-14083 62068 



•10943 71055 



296 83549 



3 23502 



11413 



219 



2 



0-56589 22342 



7985 85345 



383 02987 



6 92616 



5661 



160 



3 



1-5 



•07 



1-25407 15074 



51 71221 



3970 



1 



1-64965 05943 



99 82409 



2058 



1 



1-25327 51807 



131 31219 



3973 



1 



0-64965 09114 



99 76070 



5226 



1 



1-57 

 •00079 6 



1-25458 90267 

 2126 



1-65064 90412 

 4100 



1-25458 86999 

 5394 



0-65064 90412 

 4100 



1-57079 6 



03268 



1-25458 92393 



1-65064 94512 



1-25458 92393 



0-65064 94512 



1-57079 63268 



n 



= 



1-57079 63268 



Qn 



= 



1-25458 92393 



0n 



= 



1-65064 94512 



0n 



= 



1-25458 92393 



0n 



= 



0-65064 94512 



And here it is to be observed that the equation 0IT — 0IT = Ois (article 49, 

 equation 12) necessarily accompanied by this other, n — II = 1. 



If, in equations 1, 2, 3, 4 (article 45), we suppose t and u to be each equal 

 to the above abscissae II, we obtain for the quaternary functions of 2 II the 

 formulae 



