230 KEPORT— 1866. 



is projected as a curve east of the centre of the apparent disk. At the point 

 of mean distance the two motions coincide in value, but only momentarily 

 so, the greatest libration towards the east is attained, the orbital motion 

 becomes slower than the axial, and the first meridian returns westwardly, 

 attaining its mean position at the passage of the apogee. In consequence of 

 the small difference of the period of the revolution of the apsides and half 

 that of the nodes, the equator will not appear as a straight line across the 

 apparent disk, when the first meridian returns to its mean position, and 

 therefore the point of 0° latitude will not be found at the centre of the appa- 

 rent disk ; the divergence will be greater at the end of every period either of 

 the passage of the nodes or apsides, increasing for a period of about eighteen 

 months, after which the divergence will decrease during another period of 

 eighteen months, and at the end of three years (nearly) the state of mean 

 libration wiU be again attained. 



Libration in longitude from apogee to perigee is the opposite to that above 

 described, from which it foUows that libration in longitude changes sign 

 every lunation. 



21. The mathematical portion of this investigation may be treated under 

 two heads, viz., the method adopted in the ' Nautical Almanac,' and that 

 adopted by Lohrmann. Eor the method adopted in the ' Nautical Almanac ' 

 we again refer to fig. 5, the reasoning being as follows : — 



Since I is a very small angle, the equation tan (A' — ?3) = tan (X — S) 

 sec. T (see section 15) gives by a known formula of expansion A'=\ + sin 



2(X— ?3)tan^ — , the rest of the terms being insignificant. The second 

 term is A \ in the ' Nautical Almanac' Because T is very small, and 



COS I 



B' is always less than I, sin or ^, will be very nearly = to unity. Also 



because I — A', (p^, and /3 — B' are all small arcs, we may substitute the arcs 

 for theii' tangents and sines. Hence 



Z-A'=cos 9 (/3-B)=«' (/3-B') and ^^=/3-B'; 



consequently ?=\ + sin 2(\— ?3 ) tan^ r- + — 



^ J- 



It' 

 = A+A\ + ^; 



and as the libration in longitude Z' = ? — Z^, where l^^\he moon's mean lon- 

 gitude, the libration in longitude=X-|- A X + p — Z^; but since, as mentioned 



■^ . 

 in section 9, — h' is to be substituted for tp^, the expression becomes 



Z'=X-|- A X— — — ?„, as in p. X of the ' Nautical Almanac' 



a' 



22. Lohrmann, whose symbol for the moon's mean longitude is l,*and 



for the libration in longitude is 1', gives, in ' Topo§|^phie derSichtbaren 



Mondoberfliiche,' p. 28, the following formula for computing the libration in 



longitude: r=L — L' (see section 18 and fig. 6). NowL=270°-|-B— A and 



