ESSAYS 



THE OPERATIVE BOOTS OF THE CIRCLE-FUNCTION 



(Sir Ronald Ross). 



1*0. Nearly two hundred years ago, in the first pages of his Miscellanea 

 Analytica (London, 1730), A. De Moivre pubHshed the famous theorem with 

 which his name is associated, and which is now commonly written 



{cos6 + sind .V - i)" = cosnd + sinnd .V- i. (i) 



In Science Progress, vol. xv., No. 50, p. 628, April 1921,1 I showed 

 that this theorem is only a particular case of a wider operative theorem 

 which may be written, in the notation used by me. 



[cos 6 .0 + sind . Vr^ - O^]** = cosnd .0 + sin nd . Vr^ - 0\ (2) 



whatever real value n may have ; and I now proceed to indicate some of 

 the immediate consequences of the rule. 



This equation means that the operation contained within square brackets 

 on the left, when raised to the «th operative power (that is, when it is iterated 

 on itself w — i times), becomes the operation on the right. Now, Vr^ — O* 

 is the operation which constructs a circle of r radius about the origin of 

 rectangular axes. Denote it by Kr : then if we use the proper measure 

 for angles — what I call Quadrantal Measure — we have 



K^ = [Vr^ - 02]*» = cosm.O + sin m . Kr. (3) 



That is, if m be a fraction, the expression on the right is an wth operative 

 root of the circle-function. If we now put r = o and substitute 6 for m — as 

 the Quadrantal Measure enables us to do — we have 



Kl = [V - O^f = cos6.0 + sine.K^ (4) 



where K^^ lO. This was previously shown to be De Moivre's Theorem ; 

 since 



[{cos $ + I sin 0)O]>* = {cos 6 + i sin 5)«0. (5) 



1*1. It is easy to see that K^ is the operative ratio of two chords drawn 

 from a point on the circumference of a circle of r diameter (not radius) and 

 separated by the angle Q. That is, if a and b be the two chords, 



a^K'b or ==K;, (I) 



the double fraction-line being used to distinguish an operative ratio or fraction 



* In that note I asked for information as to whether the proposition here 

 given is already known, but I have had no affirmative reply. Mr. Laugharne- 

 Thornton, however, points out that De Moivre gave his theorem in a different 

 form to the one used here; and the Rev. J. Cullen, S.J., Stonyhurst College, 

 kindly informs me that, as I said, my proposition is not given in the works 

 of Hamilton, Tait, or Joly on Quaternions. 



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