750 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



which is rational, wherefore the sum of the two angles is muarif, and thus they 

 must belong to the same system. 



80. The irreducible surd which enters into the expression for the sines of the 

 angles of one system, may be called the modulus or mastar of that system ; we 

 shall generally denote it by the symbol \/»z. 



81. Having given the modulus of a system, to find general expressions for the 

 sines and cosines of all the angles belonging to it. 



Let \/m be the sine of any angle (/), then the expression for the cosine of that 



angle is,- 



cos 





and in order that d may be a muarif angle, it is necessary that its cosine be 

 rational, or that/'^— ?w^^ be a square number. ^ We may then put/"— w?€^=/)^, or 

 f'—f^ — me'. Now, whether e be prime or composite, we may always put e=xy, 

 when we do not exclude unit from the values of x and y; and similarly we may 

 put m = /uv with the same generalisation ; our equation of condition then becomes, 



wherefore in general, 



and 



so that 



and also, 



f+p=ixx^, f-p = vy\ 



f= 1 (fxx- + vy-') ; p = i (/^«2 _ j,^2) 



IJix^ + vy 



2wy\/ixv _^ 



/ua?^ + vy^ 



\xvx'^—y'^_ 



o „ = sin^; 2 



fxux^+y [xvx^^-y^ 



cos 



e. 



are the expressions for the sines and cosines of all muarif angles of the system 

 \/]jy, X and y being susceptible of all integer values including unit. 



It is obvious, that in assuming values for x and y, we may reject those 

 couples which have a common divisor. 



82. The sines and cosines of the sum and difference of two muarif angles, 

 determined by the above formulae, belong to the same forms. 



Let 



cos 0: 



COS e^ l^l-'y l COS 0=/i4i:i^;, 



_ 2 X yVfJiV 



Sin 



6- 



ixx^ + vy^ 



2xy\/jxu 

 fJLx'^ + vy'^' 



sin © 



then 



cos (6^ &)J-f'-^ ^ -i^yyf-fxu(xy + y x f 

 (fxx X — vyyy + fJLu (0.7 + 3/ X ) 



• //3 ^N 2 'li« X — uyy) (xy + v x.) / — 



sm (0 + 0: = , J^ v2 '^\' — ^ — S2 "VfJiU 



(ux X - vvyY + uiv (xy + yxY ^ 



