44 N. F. DUPUIS ON THE SYMBOLIC 



(8.) In a similar manner we find, — 



r0—Vcp=2ifiinl(^ft—cp) V^(H+qj). 

 (9.) LogeFi9 = log^ (cos ^+i sin 6*) log,, e =i0. 



^re-^ive; §,ve=rx, 



the differential coefficients being periodic. 



(11.) Ç Vf) dO = — i ve ; ÇÇ vo do do = - vo; ,c-c., 



the integrals being periodic. 



(12.) ve + V~^0 = Vft + y{-0) = 2 COS \{f>+ 0) V\ {0-6) = 2 cos 0. 



(13.) ve - v'e =ve- v{-e) = 2 i sin j (e+ e) vh {.e-O) = 2 * sia e. 



(14.) "When the expression \ —Ve is multiplied by 1 —V'^e, it is rendered real. Or 

 more generally, 1 — xVnH is a realizing factor for ] — x V~ nO; and reciprocally. 

 Similarly 1 -\- xV nd is a realizing factor for 1 + xVnO ; and reciprocally. 



The realized product in the first case becomes 



1 — X ( Vne + v^\e) + X-, 



which reduces to 



I - 2a;cos(n6') + a.'2. 



In the second case it reduces to 



1 + 2 .Ï cos (nd) + X-. 



(15.) The realizing factor for 1 — ixVne is 1 + ixV~\ie ,- and conversely. The realized 

 product is l + 2a; sin (w6') + :r''. 



^ Similarly, the realizing factor for 1 + ixVnd is 1 — ixV~\e, and the realized product is 



1 — 2x sin (nO) + .x°. 



^:^ m 1 + I Fé» . , 



l!iX. : io express . _ „^ with a real denominator. 



\ + ive 1 + i v'e 2t cos e i cos o 



l-iVe i + iV-'e 2 + 28in6' l+sin^ 



II. 

 Ex. 1. To factorize x"—l with u even. 



Equating to zero gives x" = I = V2k7r (3.) 



,; 21c TC 



. • . x = V 2k7t = V , 



n 



where k is to take all integral positive values from to n — 1 inclusive. 



