AS EXHIBITING THE ROTATION OF THE EARTH. 25 



The pendulum will not move in a plane passing through the vertical, but on a 

 conical surface differing slightly from such a plane, and there will ensue a slight 

 apsidal motion reverse to, and diminishing, the apparent horary motion. 



(l>) The equation ™^ = — is usually solved by the aid of elliptic in- 



^^ ^ (It Z^sin-e /■-— 2- ^ ^ ^ 



tegrals {vide Prof. Peirce's Analyt. Mech., p. 418); but for present objects the 



ordinary processes of integration are preferable. 



From equations (8)2 and (4) may easily be deduced, neglecting terms containing 



V, and bearing in mind that 



x'^-\-y"^^l- — Z-, X dx-\-y dij^ — z dz, 



— Zc?z 

 (e) di= 7 — , 



in which the upper or lower sign of the radical is to be taken according as dz is 

 positive or negative, that is, as the pendulum is descending or ascending. The 

 quantity under the radical may be put in the form {vide Mec. Celeste, Bowditch, 

 Vol. I. p. o2). 



2g{a~z){z-h)(zJ^Q 



in which 



2g~ a +77 

 (,. r..J^g i^-cr) (P-b^ 



_ (t- — or — ab — b-) 



a and b being the greatest and least values of z. 



If now we transfer the origin of co-ordinates to the lowest point of the spherical 

 surface by substituting for z, a, and b, l—v, 1 — a, I — /3, and replace C and dt in (Z>) 



f 

 by the values above found, we shall have (putting /-[-■ =zp) 



d^ 







«+6 v{2l—n) [(p—n) («— a) (/i— ?01^ 



the varying sign of the radical being understood. If we develop the two factors 

 (21 — w)-' and (p — u)~', and multiply the results, we shall ha\c 



,7^_ kP-o.')(l'^^)f bhi r I I /i I l^t 



Strike out the common factor 7, and remove the factor 2p= into the denominator 

 of the first radical factor (which factor then becomes ^(/_a) (l—b)=Va[3), ^nd 

 the above integral becomes {vide Hirsch, Integral Tables, pp. 160-164), writing V 

 for — a/3+(rx+/5)H — u\ 



4 January, 1373. 



