628 SCIENCE PROGRESS 



The expression 



f'^ — {cos 6 .X -\- sin 6- yfv"^ — x'^Y 



is easily seen lo be a complete square if we put cos'^Q .{y- — {r" — x"^)] for 

 cos:^6 . x' ; that is, it equals either 



{cos 6 . Vr^ — x^— sin 6 . x)^ or {sin 0.x — cos 6. Vr^ — x'^)'^. 



Now if y = cosd .X + sin 6 . Vr^ — x^ and x = cos cf) . z -{- sin (f) . Vr^ — 2", 

 it follows that (taking the first of the two squares) 



y = cos6 . {cos (f> . z + sin (f) . -y/r^ — ^^J + si n 6 . {cos (p . \/r^ — z^ — sincf). 2) 

 = cos{d + (f)) .z -{■ sin {6 + (f)) . y/r"^ — z^. 



This process can evidently be continued indefinitely, and, if all the angles 

 ^, ^, . . . be the same, is a case of iteration, or successive substitutions, of 

 the same operation. Thus, if the original function cos6 .x + sinQ . ^/r^ — x"^ 

 be iterated {n — 1) times on itself the result will be 



cos nd . V + sin n 6 . Vr^~—v^ ; 



and this can easily be proved to hold for any real values of n. The proposition 

 may be exhibited in a single equation by means of my " explicit operative 

 notation," which I employed and explained in the previous paper mentioned — 



[cos e.O + sine . ■s/y'^ — o-]" = cos «(9 . o + sin nd . Vr'^ — O^. 



(Here O is the " symbol of substitution," and the index n affixed outside 

 square brackets denotes operative as distinct from algebraic involution.) 



Now, so far as the mechanical process of algebra can inform us, this pro- 

 position holds true whatever value r may have — that is, the mechanical 

 result is independent of the numerical magnitude of r, while O has no 

 numerical significance at all. H ence the same result will be obtained when 

 r is zero. In this case Vr^ — O^ becomes V — 0=*, which = O . V — i ; and 

 our equation becomes 



[{cosd + V — I . sine)o]^ = {cosnd + V — i . sinn6)0. 



But the iteration of any function a . o, where a is a constant, is given by 

 [ao]'» = a'».0 (that is, ax iterated (n — i) times becomes a^x). Hence 

 finally, 



[{cos 6 + V — I . sin (9)0]" = {cos 6 . + V — i . sin 6)^0. 

 and {cos 6 + V — i . sin 6)» = cosnd + V — i . sin n 6 ; 



which is De Moivre's Theorem. 



This proposition has so many implications, especially as regards vectors, 

 complex numbers, and trigonometrical functions, that I shall be very grateful 

 for any information regarding it. In fact, these functions are connected with 

 the wth operative roots, or with the operative logarithms, of the primary algebraic 

 operations ; and the imaginary unit seems to be only the symbol of error which 

 occurs when the operative mechanism is wrongly applied to numerical 

 subjects. 



Sociological Society 



A meeting of the Sociological Society was held at Leplay House, 65 

 Belgrave Road, on Tuesday, January 25, at which Mr. H. J. Laski spoke on 

 " The Prospects of Parhamentary Government " ; Prof. Graham Wallas in 

 the chair. 



Mr. Laski said that Parhamentary Government in this country had been 



