736 Sir G. Greenhill on Pendulum 



ARD is 7r ; the tangent intercept ZZ' by the cyclic arcs 

 is \it, so that QZZ' is a spherical triangular octant ; 

 RZ=ZK, RD = DK, RA = AK' ; and if R, Z moves to r, s, 

 as Q advances to q, and Dz crosses ZR in z\ 



(1) cos c du = cos DQd?</> = sin DZd<j> = Zz / 



= arc Rr— rz + RZ, 



(2) u cos c = arcER— RZ, 



thus representing the time by the difference of the arc Ell 

 and RZ. 



*tt> In the reciprocal part of the diagram in fig. 5, where the 

 tangents at R, R x intersect in S, 



(3) AS + DS-tt-/, 



(4) (w!-m)oosc= arcRRi— RiZi + RZ; 



(5) arcRR 1 = SR + SR 1 - constant (Salmon, § 252), 



(6) (M 1 -w)cosc+aconstant=SZ-SZ 1 =8Z + SZ 1 , -i7r. 



The Spherical Trigonometry interpretation is the same as 

 before in §8; since 



(7) 2 = XQ=XSQ=i7r-DSZ = i7r-ASZ/, 

 sin q = sin DQ sin QDX = k sin </>' sin yjr, 



, ftN or7 cosZDS cos^ir 



< 8) cos feZ = -.~- T ^r/ = 1 



v y sin JJfeZ cos ^ 



. or7 sinZDSsinDZ sind-Ad/ 



sinfeZ = : — p-^7 = T - 



sm JJoZ cos q ' 



/0 x ««/" cosZ/AS cos ft' 



(9) C0SPZ X as . A Qf7/ =- —j 



v y sm AbZ/ cos ^ 



• Q 7 /_ s i n Z/AS sin AZ/ sin (j>'Ayjr 



sm fezJi — ; — . r/ . = - 



sm ASZ/ cos q ' 



(10) sin (SZ + SZ.Q = Si " ± C0S *' A *' + Si " ^ C0S ^ A ^ 



y cos 2 </ 



= sn (K — w + ^i)=sn (K + wj), 

 SZ + SZ/ = am(K + ™). 



