118 Mr DAVIES on the Nature of the Hour-Lines 



tions of the angles, the functions of which enter into the formula just given. 

 The value of L in that case would be 



1 - 1 f 1 1Q 4Q' <>0" 1 



L = sin \ cos 109 49' 50" cosec J y I 



n ^ n n J 



and the only difference is, that the errors (the same in value) are affected 

 by the contrary sign. 



It has been remarked that the results given by the formula above are 

 always greater than the error arising from the substitution of great circles 

 for the true hectemoria. A few simple considerations will establish this. 



Let the plane of that great circle on the sphere which joins the equi- 

 noctial and tropical hectemorial points be produced to cut the equatorial cy- 

 linder. This cylinder being again developed, the elliptic section thus made 

 becomes a new harmonic curve. We have to prove that this new harmonic 

 curve passes between the former and the chord of its developed intertropical 

 segment. 



The harmonic curve is always concave to its axis, and hence all the 

 chords, which lie wholly on one side of the axis, lie wholly within the curve. 

 Now, both these harmonic curves are referred to the same axis, and cut one 

 another, and hence the common chord must fall within them both ; or, 

 which is the same thing, they both lie on the same side of the common 

 chord. Evidently, then, that which has the least curvature must fall be- 

 tween the other and the common chord. But the extreme values of y be- 

 ing the same, that curve will have the least curvature whose range of ab- 

 scissa is the greater, or, which is the same thing, whose generating circum- 

 ference, and consequently generating radius, is the greater. Hence the new 

 harmonic curve falls between the other curve and the common chord ; and, 

 therefore, if we restore the developed cylindrical surface to its former equa- 

 torial position, the great circle which forms the spherical chord of the inter- 

 tropical hectemoria will fall between that in hectemoria and the line which 

 formed the rectilinear chord of the harmonic curve. The abscissal distance 

 between the great circle chord and the true hectemoria, for any specified de- 



