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III. 



VERB-FUXCTIOXS, WITH Ts^OTES O^s" THE SOLUTIOjS^ OF 

 EQUATIOJs^S EY OPERATIVE DIVISION. 



By RONALD ROSS, D.Sc, E.R.S., C.B. 



Eead February 13. Ordered for Publication Febrlarv 15. 

 PubHshed April 6, 1905. 



I. Inteoductiox. 



1. In all branches of mathematics the need is felt for an algorithm 

 capable of rendering algebraic operations apart from their subject and 

 at the same time in a manner which will express their exact con- 

 struction. For example, il y = a-{-hx,^e understand that an operation 

 has been performed on x which has converted it into y ; and we can 

 state this idea implicitly by writing y=<^ {x). But when we endeavour 

 to represent ^ — that is, the operation itself apart from its subject — 

 explicitly in terms of the coefficients a and h, we find ourselves at a 

 loss how to do so. We cannot equate <^ to anything. We cannot 

 write (fi = a b, or cf^ = a + ix : this would be to equate an operation 

 to a quantity — a verb to a noun. In fact, we can only infer the nature 

 of (f> by observing the effect which it produces on the subject. The 

 result is a limitation of our powers of expression ; we can easily 

 represent explicitly the relations of quantities, but not so easily those 

 of operations. For example, if cj) = a + bij/ + ex^, or it cf> = if/x, we 

 know that these relations hold between the operations cf), if/, and x 5 

 but when we wish to exhibit the structure of the operations simul- 

 taneously with their relations to each other, we can do so only by 

 the assistance of other equations defining each of tlie elements sepa- 

 rately — we cannot put the whole information into a single equation. 

 Nor can we easily represent repeated or inverse algebraic operations 

 without circumlocution. 



2. It may therefore serve a useful purpose to discuss a means — 

 probably the only means — of meeting this want. Any complex of 

 elements may be conceived as being the result of an operation 

 performed on that one of the elements which, for the moment, we call 



R.I. A. PROC, VOL. XXV., SEC. A.] C 



