86 Mr. W.G. Horner on the Signs of the Trigonovietncal Lines, 



ring being placed, the pressure of the screw E, forces it 

 against the two uprights, which tightens it, and by that 

 means fixes the wedge D in figure 2. 

 Fig. 4. Ring and stand complete for experimenting with. 



Fig. 3. 



Fig. 4. 



XIII. On the Signs of the Trigonometrical Lines, 

 By W. G. Horner, Esq,^ 



1. "D EGARDED as a demonstration of the algebraic af- 

 ■^^ fections of the trigonometrical lines, the following state- 

 ment (it is hoped) will be found to combine graphical distinct- 

 ness with mathematical evidence. But its chief presumed re- 

 commendation is its completeness. In every treatise that has 

 fallen in my way, the affections of the chords have either been 

 overlooked, or imperfectly, not to say erroneously, discussed. 

 To remedy this fault, the entire system of chords is intro- 

 duced into the annexed diagrams, and the chord is made a 

 principal element in the investigation. 



2, The lines C A, AT, (fig. 1.) being at right angles to each 

 other, about the points C and A let two indefinite right lines 

 C P, A P, revolve continuously in the direction ^''"^ . At 

 the outset let the former coincide with C A, and the latter 

 with A T ; and let the latter revolve with half the angular ve- 

 locity of the former. Then [a] when C P next coincides with 

 C A, A P will coincide with T A produced in the direction 

 averse from T ; for this is only saying that the former will 



♦ Communicated by the Author. 



