ON MAPPING THE SURFACE OF THE MOON. 



225 



by a small angle found subsequently. These substitutions being made in the 

 formulae for sin C, and changing C into C, we have 



(?-?8+A) 



sin C= — sill i cos 



cos 



sin 



. cos(a'— £3') 



1 71 '^ 



cos b 



which are the formulae on p. s of the * Nautical Almanac ' for computing the 

 angle C. 



10. The angle C changes sign with cos («'— 8 ') and i, the change of sign 

 of i being due to the motion of the moon's nodes. It does not change sign 

 with the changes of sign of I' and V . It is positive when the northern part 

 of the circle of declination is to the west of the moon's meridian. 



11. In fig. 3, from Lohi-mann, we have MP, the moon's pole; EP the 



Fig. 3. 



90 i 



IBOi 



■■4o T- 



2,10° 



earth's pole ; M the centre of the apparent disk ; (MP) (E P)=i the inclina- 

 tion of the moon's equator to the earth's equator ; (E P) M=p', the N.P.D. of 

 the moon's apparent centre 90° + ^'; (MP) M=a, the distance of the moon's 

 apparent centre from the moon's pole=P cr, fig. 2, =90° — ^i ; the angle 

 (MP) (EP) M=A the inclination of i to jy=90°+a'— £3' (see section 8), 

 or 270°+ £3' — a' (see Lohi-mann, 'Topographic der Sichtbaren Mondobcr- 

 fljiche,' p. 27) ; the angle (E P) (M P) M=B the inclination of i to a=90°— 

 Q,— ?3 + A) (see section 8) ; the angle (E P) M (M P) = C the inclination of 

 p' to a (see section 8, angle P(r7r=C'). 



The formulae for computing the angle A and the sides i and ^' are given 

 above. The Gaussian formulae for obtaining the values of B and C, with 

 the side a are as follows : — 



t„„ 1 /-n p^ cos|Asin|(2y-i) (1) 



tan g (B-L)=g^ | A sin | (^Z + i) (2) 



tan 1 (V.^V\ cos I ^ COS I (/-^) (3) 



*^2^^ + ^^-sin|Acosi(2y + i) (4) 



B=i(B + C) + i(B-C) C=i(B + C)-i(B-C), 



sinia='iii4A!^^iii!l±i!. 

 ^ cosi(B-C) 



1866. Q 



