30 Proceedings of the Royal Irish Academy. 



Again by consideration of the variation of the argument of any rational 

 integral function of s of degree n, when the point which represents s by 

 the numbers giving the lengths of its rectangular coordinates is made to 

 describe a closed curve, we can prove that the function may be expressed 

 and uniquely expressed in the form (s + •/,) (s 4- -y.) : - - _ >, &c, or thrown 

 into the form of n factors. So any order z" ■*■ «i="~' -r a^z"'" + &c, with 

 respect to the distribution of water in the four vessels can be uniquely 

 carried out by n consecutive orders s + y u z + y., &c, when z is arbitrary. 



XOTE. 



The conception of y/- 1 as a geometrical operator signifying rotation 

 through a right angle was introduced by Argand in 1S06. I have shown 

 that it is not necessary to restrict the angle to a right angle, and note that 

 such a conception of ^/ - 1 is limited to geometry. Finding other modes 

 of introducing i unsatisfactory — for instance, in Harnack's Introduction to 

 the Calculus, " i is called the imaginary unit and operations with it are 

 defined" — I have ventured in this paper to show that numbers both real 

 and complex have a practical concrete meaning, hoping to help mathe- 

 maticians who have no inclination for the study of abstract logic to satisfy 

 themselves that the processes they make use of have a definite meaning. 

 Those who prefer a more abstract treatment of the subject may be referred 

 to Bertrand Russell's " Principles of Mathematics," vol i (Cam. Univ. Press, 

 1903). Those who are satisfied with the treatment of the real variable may 

 in most simple processes, at all events with the complex variable, derive the 

 required result by treating i as a variable parameter : see " The Hypothetic 

 Variable," by R. A. P. Rogers, Hermathma, vol. xiv, 1906. I regard in 

 this paper all numbers as operators on a fundamental unit, but I also have 

 ventured to regard them as adjectives, because ordinary numbers are such 

 grammatically. When we speak of 5 tons of hay, 5 is an adjective. When 

 we further qualify the 5 tons by specifying whether it is hay that has been 

 bought or sold, the symbol which signifies all this is an adjective. When 

 we further extend the relations of a ton of hay. I still call the operator 

 . - iv a complex adjective, following the same extension of terminology 

 aeeording to which we call it a complex number. 



