Prof. Maiden on Cosines and Sines of Multiple Arcs. 1 1 3 



ance to conduction be great, as it most probably is when po- 

 tassium is slowly converted into potassa by the action of a 

 mixture of oxygen and common air ; or little, as it probably is 

 when a mixture of oxygen and hydrogen is exploded ; still the 

 quantity of heat evolved remains proportional to the number 

 of equivalents which have been consumed, and the intensity of 

 their affinity for gaseous oxygen. 



64?. That the heat evolved by other chemical actions, besides 

 that which is called combustion, is caused by resistance to elec- 

 tric conduction, I have no doubt. I cannot, however, enter 

 in the present paper upon the experimental proof of the fact. 



Broom Hill, Pendlebury, near Manchester, 

 October 5, 1841. 



XVIII. On the Development of the Cosines and. Sines of Mul- 

 tiple Arcs. By Henuy Malden, M.A., late Fellow of 

 Trinity College, Cambridge, and Professor of Greek in Uni- 

 versity College, London *. 



IN the Philosophical Magazine for August 1841, [S. 3. vol. 

 xix.] Mr. Booth has developed the cosine of a multiple arc 

 in descending powers of the cosine of the simple arc, by the 

 application of a new theorem in the calculus of finite differ- 

 ences, and has demonstrated that cos 7i 6 cannot be so ex- 

 panded when n is either negative or fractional. 



As this is the case, it seems sufficient to present the expan- 

 sion of the integral multiple in an easier way, 



cos {71 + \)6 = 2 cos 6 . cos nO — cos {n — \) 6 : 

 cos n 6 cos n 6 



hence 



cos [n + \) 6 2 cos 6. cos n6 — cos [n — I) 



1 



~ _ a cos {n — I) 6' 

 2 cos U — 



cosn 6 



and as the fractional term in the denominator is of the same 

 form as the fraction on the first side of the equation, this may 

 be developed in the form of a continued fraction, thus: 

 cos n _ 1 



cos (ti'+'iYe ~ r~o ^ '' 



2 cos ^ — 



2 cos ^ ^ 



2 cos ^ - &c. 



in which 2 cos 6 recurs fi times as the integral part of the par- 

 tial denominators, and tlie last fractional part is 

 cos ^ 1 



cos a cos a 



* Coinnuinicated by the Author. 

 Phil. Mag. S. 3. Vol. 20. No. 129. Feb. 1842. 



