ii8 SCIENCE PROGRESS 



teen years ago I tried to expose the fallacy and to suggest the proper solution 

 in a lengthy paper (2). The truth is that the " identical operation " <^° 

 cannot possibly be equated to numerical unity or to any number, but is 

 itself the unit of operation. Denote this by O for brevity, and set aside a 

 special operative bracket (the square bracket is the best), and we possess at 

 once a powerful addition to our mathematical algorithm, which enables us to 

 apply either algebraic or operative processes to the same equations, at will 

 and with equal rigidity. It is not symbolism, but merely the logical com- 

 pletion of the ordinary algebraic and functional notation now in general use.^ 



3'0. It is advisable to begin with the simplest algebraic proof of the 

 proposition 1-02. Perhaps the most useful one is the following. Let 



y = cos y . X + sin y . Vr^ — x^, x = cos 6.2 + sin 6 . Vr^ — z^ ; 



that is, 3/= [cosy . o + siny . Vr^ — 0^]x, x= [cos6.0 + sind. Vr^ — 0^]2; 



and thereiore y =[cosy. O -{- siny. V r^ — O'^][cosd .0 + sin 6 . Vr^ — 0'^]2. (i) 



Now, it can easily be shown (see 3-61) that 



r- - {cos e.O + sin e . Vr^ - O^)^ = {cos 6 . Vr^ - O^ - sin 6 . Of. (2) 



Hence [cos 7.0 + sin y . Vr^ — O^] [cos 6.0 + sin 6 . V>^ — O^] = 



= cos y {cos 6.0 + sin 6 . Vr^ — O^) ± sin y {cos 6 . Vr^ — O^ — sin 6. o) 

 = cos {Q ±y) . O + sin {d ±y) . Vr^ - O^. (3) 



This process can be continued for any number of angles, and the reason for 

 the ambiguity of sign will be explained later (4*26). Now write 



Kr^+Vr^-0\ (4) 



and suppose that 6 is -th of a right angle, so that, in Quadrantal Measure, 



cosi . o + sin 1 . Kr= Kr: 



= cosd .0 + sin 6 . Kr. (5) 



Hence if we use only the positive value of the radical, 



KlKl = Ky=^KlKl: (6) 



that is, these operations are Abelian, or permutable. 



It can easily be shown by the methods employed for De Moivre's Theorem 

 that n, and therefore 6, may be positive or negative, integral or fractional. 



3*1. If we use either value of the radical for Kr, we obtain ambiguous 

 results which are inconvenient at present for deducing its properties. This 

 can most easily be done from the known properties of the trigonometrical 



1 Thus when we wish to render an operation explicitly but apart from its 

 subject, we replace that subject by <^°, or O ; and when the expression so 

 obtained is to operate on the following matter (as (/> operates on x in (px), 

 we enclose it in square brackets ; and when it operates on itself n — 1 times 

 (as <p does in c^"), we affix the exponent n to the square brackets. On the other 

 hand any number or expression in other than square brackets is merely 

 multiplied into the following matter (as in (0)^) ; or is raised to its «th 

 algebraic power by an exponent (as in (0)**). The powers of O, such as 0», 

 are algebraic ; and ^ . yj/ denotes the algebraic product of (p and yjr. 



