Logarithmic Lazy tongs and Lattice-works. 377 



and to the first order of b/a, 



. 2 Pa 3 /. 7 Pa 2 k 2 <? \ 



is=—» 1 -^--i--— -cos- cos 6^"^ 



r 3tt r \ 2d 10 / 



h 2 a 2 b/. , 8 cos fl\ , , &V6/ 2 76 cos 6 1 9n 4 3/l \ 

 or \ 7r 2 J r \ 4o /57t 2 90 37r 2 / 



, . v . . (48) 



When the incidence is oblique the analysis is much more 

 cumbrous, and there is little interest in carrying it beyond 

 the approximations given by Lord Rayleigh in the paper 

 cited above. 



XXXIII. Logarithmic Lazytongs and Lattice-works. 

 By Thomas H. Blakesley *. 



THE point o£ view in which the Equiangular Spiral is 

 usually regarded is that implied in its name, viz., the 

 curve which makes the same angle with its radius vector, 



d6 

 r-r = tana. 

 dr 



It is rather from what I may perhaps call its polygonal 

 character that I shall present and apply it in this paper. By 

 this I mean that it is a circumscribing curve to polygonal 

 figures following simple laws. 



If a series of equal straight lines form a consecutive 

 number of the sides of a regular polygon, the circumscribing 

 circle is absolutely determined. 



But if those straight lines, still maintaining the equality of 

 the angles between any consecutive two, in magnitude form 

 a geometrical series, the circumscribing curve will be the 

 equiangular spiral. 



According to the value of the angle between consecutive 

 lines, one may speak of the figure as a regular logarithmic 

 pentagon, hexagon, octagon, &c, and more generally as a 

 regular logarithmic polygon. The regularity consists in the 

 equality of the angles between the lines, and in those sub- 

 tended by them at the pole of the spiral. 



Consecutive chords are those straight lines which form 

 consecutive sides of a logarithmic polygon. 



* Communicated by the Physical Society : read March 22, 1907. 



