188Q.] Mr Glaisher, On some theorems in Trigonometry. 383 



the triangle ABC, and then make the angle CAB turn about A, 

 and the angle AB C about B ; and let C denote the point of con- 

 course initially at C, and K the point of concourse of the other two 

 arms of the rotating angles. Now let C" coincide successively 

 with DEFO, and let the corresponding positions of K be PQRS. 

 Describe the conic APQRS by the organic method, and let K 

 move on the conic thus described ; then C describes the cubic as 

 required. See § xxxiil. p. [159]. 



4. Newton further remarks that curves of the higher orders, 

 having a double point, may be described in like manner. He also 

 points out that one finite angle and a vanishing angle (or straight 

 line) may be used in the Bescriptio Organica, in place of two finite 

 angles. The full development of what he has there briefly stated 

 may be found in Maclaurin's Geom,etria Organica (1720). 



(2) Mr J. W. L. Glaishek. Addition to a previous paper on 

 some theorems in trigonometry. 



The following is an addition to my paper on pp. 319 — 329. 



§ 12. Theorem. If we have, identically, 



2.nsin(a-;8)=0 (1), 



where 11 sin (a — /3) denotes a product of sines of differences of 

 angles, and S denotes the sum of any number of such products, 

 each of which however must contain the same number of sines, 

 then 



S . n sin (a - /9) sin {a + j3) = 0. 



The proof of the theorem is very simple for in virtue of (1), by 

 expanding the sines and equating the terms of lowest dimensions, 



S.n(a-;8) = 0, 



whence, writing in this identity a.^, /3^, ... ior a, fi, ... 



X . n (a^ - /3^) = 0, 



and replacing a, /3, ... by sin a, sin /3, ... the identity becomes 



X,n(sin'a-sin=^yS) = 0, 



that is S . n sin (a — /3) sin (a+ ^) = 0, 



§ 13. As an example, take the well-known identity 

 sin (;8 — 7) sin (a — S) 

 + sin (7 — a) sin (/3 — S) 



+ sin (c« - /S) sin (7 - 8) = (2); 



VOL. III. PT, VIII. 28 



