Section III, 1891. [ 43 ] Trans. Roy. Soc. Canada. 



VI. — On the Symbolic Use of Demoivres Function. 

 By Prof N. F. DuPUis, Queen's University, Kingston. 



(Read irfay 29, 1891.) 



Tlie contents of this paper are as follows : — 



I. The properties and laws of transformation of the operativ'e symbol VO where 

 F is a contraction for Demoivre's function, — 



Cos e + i un 0. 



II. Applications of the operator V in the summation of certain trigonometric series, 

 and the expansion of certain functions. 



(1.) "We define VO to be such a function of that ViiB = ( VB)" ; or writing the right- 

 hand member after the form usually adopted for trigonometric ratios, VnO =: V"t), where 

 n and 6 are any quantities whatever. 



We know that this relation is satisfied by Demoivre's function, cos fl + « sin 0, and 

 hence that Vd = cos 6 + i sin H. 



(2.) Siuce V(r-ft) = cos {—6) + i sin (—0) = cos .0 — i sin «, we have F(— ^) = V-^B. 



(3.) F(2;t7r) =Cos 2k7t + i sin 2kTt — Vln = + 1, 



r(2A+l),T = Cos (2/c+l)7r + «sin (2/t + l);r= Yn = —\; 



where k is any positive integer. 

 (4.) Since VnB=.V"d= t\, 



. ■ . VnB X Vacp — V^n x F^n = F«(^+ cp) = V(nB+n(p). 



Or making m = 1, 



re- Vcp = V{B+cp); 

 and VB ■ F(— y.) = V(B—cp). 



(5.) F(7r+ H)=r7r- VB= -1- rB = — VB, 



by the second part of (3.) 



(0.) VB =V6-VB= (cos B+i sin 8) VB 



= cos 8 (cos B + VB+i sin ^)— 1 

 = 2 cos ^ F(9— 1 



Addition Theorem. 



(7.) YB-\- Vcp = cos B + cos <p + i (sinl9 + sinç») 



= 2 cos ^ ('9+9') cos J (6"—^) + 2( sin J ((9+ip) cos ^ {B—cp). 

 = 2 cos J (^— ç») Fi («+<^) 



