and the Methods of calculating their Results. 273 



not assumed as absolutely known, but corrections are applied 

 to the assumed values of the same, so that a + A a, S + AS 

 ...denote their true values. If these corrections are supposed 

 to be so small that their squares, products, &c. may be neg- 

 lected in the alterations which P — u and Q — v will undergo 

 by the substitution of the corrected values, the question is to 

 find, by means of the equation 



(*)...(£+ Ak)'= [P-u + aAa + bA$ + c An +dAe*Y 



+ [Q-^-f«'A« + #A8 + c'A7r + d'A ^] 9 , 



the time of the first meridian which corresponds to the time 

 of the observation, or, if the latter be denoted by t, to deter- 

 mine t — d by an expression involving A«, AS, &c. or, which 

 is the same thing, to find the relation between d, A a, A 8, 

 &c. which results from the observation. The coefficients 

 a, b ... a', b' are the differential quotients of P— u and Q— v 

 in relation to the elements of the calculation «, 8. 



[5.] Let us suppose that the required time of the first me- 

 ridian, viz. the meridian whose time was employed for cal- 

 culating a, 8 ... consists of two parts T and T', and let P be 

 = P + p'T' ; Q = q + q' T', where p and q denote the values 

 of P and Q for the time T. If the variations of P and Q 

 were proportional to the time, p' and q' would be constants ; 

 but in reality they depend on the second and higher differ- 

 ences of P and Q, which, however, if T' does not exceed some 

 hours, have only such a small effect that the variations of p f 

 and q', compared with the variations of P and Q themselves, 

 may be considered as quantities of a higher order. On this 

 circumstance rests the solution of the equation (4) by successive 

 approximations which rapidly converge to the truth. If we 

 introduce in place of the corrections of the elements two new 

 quantities dependent on them, so that 



p'.i — q'.i' = a. Aa -f b. AS + c . An + d. Ae* 

 q\i +p'. i' = a'.Au + b' . AS + c' A ?r + d! Ae 9 

 and likewise m sin M = p — u n sin N = p' 



m cos M = q—v n cos N = q' 



the equation (4) will assume this form : 

 ( A'+ A A)*=[m . cos (M— N) +■ n(V + i)J + \_m sin (M— N)— n . i']*, 

 and will give, neglecting the squares of i' and A^ and intro- 

 ducing the auxiliary angle \J/ determined by this equation 

 k cos <J/ = m sin (M — N), this result: 



rp f m . cos (M — N + ^) . i Ak 



n . cos -4> ' tang 4* n sin -^ 



where, if \J/ be supposed < 180°, the upper sign belongs to an 

 N. S. Vol. 8. No. 46. Oct. 1 830. 2 N immersion, 



