DIALLING. 



707 





Of the line 

 described 

 on a plane 

 by the ex- 

 tremity of 

 a shadow. 



Plate 

 CCXXIX. 

 Fig. 13. 





sun shines on the plane, and the hour angle HPS, in 

 time, is the interval between noon and the sun's leav- 

 ing the plane. Taking the example of art. 73, it will 

 be found that when the sun is in the northern tro- 

 pic; and consequently PS=66 C 30', the angle SPA= 

 S2° 12'. Now we found, (art 74.) that HPA=25°20'; 

 therefore, when the sun sets upon the plane, SPH, the 

 horary angle from noon is 107° 32' = 7 h 10" 1 , the sum 

 of the two arcs ; and when it rises, the horary angle is 

 56° 52' = 3 h 47 m , their difference. Hence it appears, 

 that it will be needless to trace upon the dial any hour 

 line earlier than III in the morning, or later than 

 VIII in the evening. 



Of the Line described on a Plane by the extremity of a 

 Shadow. 



77. Sometimes we see described on dials the path 

 described by the extremity of the shadow of the axis, in 

 the course of the day, at certain times of the year, and 

 in particular at the times when the sun enters the dif- 

 ferent signs of the zodiac. As this is an interesting- 

 part of the theory, we shall investigate the nature of 

 the path of the shadow, and shew how any number of 

 points in it may be found. 



Let O, (Fig. 13.) be the centre of a horizontal dial; 

 OA the meridian line; OF the axis; and OB the sha- 

 dow of the axis at any time. From F, the top of the 

 axis, draw FD perpendicular to OB, and DA perpendi- 

 cular to OB, meeting the meridian in A, and join FB, 

 FA. Because BD is perpendicular to the two lines 

 DA, DF, it is perpendicular to the plane of the triangle 

 FDA, (Geometry,) therefore any plane passing along 

 BDis perpendicular to the plane of the triangle FDA, and 

 consequently the triangle FDA is perpendicular to the 

 plane of the dial ; but the triangle FAO is also perpen- 

 dicular to the same plane ; therefore CA, the common 

 section of the two triangles, is perpendicular to the plane 

 of the dial. 



Let a=OF, the length of the axis; 

 r=OB, the length of its shadow ; 



„ J the angle BOA contained by the shadow 



" \ and the meridian ; 

 _ f the angle FOB, contained by the shadow 

 v ~ I and the axis ; 

 L=the angle FOA, the latitude ; 

 » f the angle OFB, the sun's distance from the 

 d =\ pole. 

 Then, in the triangles FDO, ADO, both right angled 

 at D, we have, by Trigonometry, 



OD:OF::cos. FOD: rad.; 

 OA : OD : : rad. : cos. AOD ; 



therefore, ex cequali OA : OF : : cos. FOD : cos. AOD ; 

 but in the triangle OAF, OA : OF : : cos. FOA : rad. 

 therefore cos. FOA : rad. : : cos. FOD : cos. AOD. 

 From the triangle OFB we get this other proportion, 



Sin. (F + O) or sin. B : sin. F : : OF : OB. 

 By substituting for the lines and angles in these two 

 last proportions their symbols, we have, 

 Cos. L : rad. : : cos. v : cos. C 

 Sin. (5-J-e) : sin. "i::a:r 

 and hence, 



__Cos. L. cos. C a sin. J 



rad. sin. (l-j-j;) 



By these formulas, having given the sun's declination, 

 we can readily find Or, the length of the shadow, for 

 any value of C ; or combining them with the formula, 

 of art 39, viz. tan. C=sin. L tan. E, where E is the 



horary angle from noon, we can find the length of the 

 shadow at any time of the day. 



78. These two formulas may be reduced to one, by 

 elementing the angle v; for, supposing rad.zrl, we 

 have sin. v=:i/(l — cos. 2 L cos. 2 C) ; and by the arith- 

 metic of sines, sin. (S+«)=sin. 2 cos. u-J-cos. 2 sin. v. 

 By substituting for sin. v and cos. v then* values, and 

 putting sin. § cotan. ? for cos. 2, we get from the second 

 formula, 



a 



y— . 



Cos. L cos. C-f cot. d \/ (1 — cos. 41 L cos. 2 C.) 



This formula exhibits the relation between the vari- 

 able angle C, and the variable line r, the length of the 

 shadow. By giving to C any number of successive 

 values, and substituting the value of S (taken from the 

 tables of the sun's declination, p. 715.) for the given day, 

 we may find any number of points in the path of the 

 extremity of the shadow. It is not so convenient, how- 

 ever, for calculation, as the formula of last article. 



79- It is easy to see that the path of the extremity 

 of the shadow must be a conic section. For the straight 

 line drawn from the sun through the top of the stile, 

 which determines the length of the shadow, manifestly 

 describes, by the diurnal motion of the sun, the surface 

 of a cone, having the axis of the dial for its axis ; and the 

 path of the shadow is a section of this cone, made by 

 the plane of the horizon. The polar equation of the 

 path, found in last article, shews also that the curve is 

 a conic section ; for let it be put under this form, 

 a — r cos. L cos. C= cot. 2,y/(r 2 — r 2 cos. 2 L Cos. 2 C). 



Now, supposing x and y to be rectangular co-ordi- 

 nates, which have their origin at O, we have r cos.C=x, 

 and r 2 =x 2 -\-y 1 ; therefore, by substituting in the equa- 

 tion, and squaring, &c. we get 



(Cot. 1 3 sin. 8 L — cos. 2 L)x 2 -f-2a cos. Lxl _ n 

 -fcot. ^y z — a 2 | — U. 



This expression may, by the arithmetic of sines, be 

 transformed to 

 — Sin. (S-j-L) sin. (S — L) x 2 -f-2« cos. L sin. 2 2 xl 



-f Cos.* Jw'-a! sin. 2 2 J ~ a 



If L=5, then sin. (J — L)=0, and the equation be- 

 longs to a parabola. 



If L is greater than 5, so that sin. (« — L) is nega- 

 tive, the equation belongs to an ellipse ; but if L is less 

 than J, so that sin. (§ — L) is positive, then the equa- 

 tion belongs to an hyperbola. In each case, the meri- 

 dian line is the transverse axis of the curve. 



If £=90 9 , so that cos. 2=0, then the term containing 

 y l vanishing, x has the same value for the whole day ; 

 which shews that the path is a straight line. 



The path of the shadow is an ellipse at any place 

 within the polar circle, on the days when the sun does 

 not set. It is a parabola at that place on the day that 

 the sun just touches the horizon at midnight. It is a 

 straight line at all places of the earth on the equinoc- 

 tial days. And in every other case it is a hyperbola. 



80. The points in which the curve crosses its axis 

 may be readily determined from its polar equation, 

 (art. 78.) by making C=0, and C= 180°. Thus, calling 

 their distances from the centre of the dial r' and r", we 

 have 



a sin. 2 a sin. 5 





sin. (3+L)' 





(3-L)' 



The first of these is the length of the shadow at noon. 

 The vertices of the curve lie on the same side of the 

 centre of the dial, when it is a hyperbola ; but on op- 

 posite sides when it is an ellipse. The other elements 

 of the curve may be discovered in like manner, ar.il 



