Ross — Verb- Functions. 



73 



Hence y = [I)' - ^ J) + ^ D\ . .']{7i + 1) x'' = x'\ 



L£ 



When the coefficients of D'^, D, J)-, . . . are functions of higher 

 orders than the first, the exponents of B in each term of the quotient 

 must be so chosen as if possible to ensure that the first term of each 

 dividend shall not be repeated in each subtrahend — which is generally 

 obtained by the aid of Leibnitz's theorem. For example (one form), 



ri)° 4- x-'By = D'- D + - R. 



^ 2^+1 Q{2x+ 1) 



Of course, B'' here denotes operative involution. 



25. Conclusion. — Time does not allow examples of functional equa- 

 tions to be given ; but enough has been said to support the view that 

 operative division affords a general and methodical way of dealing 

 with linear equations. Eeflection will suggest that this way is also 

 the natural way. We do not, so to speak, attempt to capture the 

 solution by artifice, but, setting aside the quantitative subjects, evert 

 the original operation itself, step by step, in accordance with a fixed 

 plan. It may often happen that the result of the artifice is more 

 useful to us than the result of the general method ; but this fact does 

 not necessarily diminish the value of the latter. Operative division 

 therefore affords a good preliminary example of the uses to which 

 verb-functions may be put. 



It may be noted in conclusion that the whole system of verb- 

 functions depends on recognition of the fact that does not equal 

 numerical unity. 



The writer's warm thanks are due to Professor Joly for the 

 interest which he has taken in the matter, and for his kindness in 

 reading this paper to the Royal Irish Academy. He is also indebted 

 to Professor Carey, of the University of Liverpool, for help rendered ; 

 and to Mr. Walter Stott for first applying the general method to the 

 solution of some particular equations ; and for other assistance. It is 

 due to the memory of the late Mr. R. W. H. T. Hudson to add that 

 he was one of the first to accept the validity of some of the arguments 

 used in this article. 



