193 



According to Laplace's method of computation, using an auxiliary 

 angle A, 



sin" A = cos' 4- (/3'— /3) cos t cos -x 

 sin i V = cos (i ff + I t' + A) cos (| «• + |- *--A) 

 From these formulae, we may proceed to investigate dV 

 log sin hel. lat. = log sin geo. lat. — log sin STC + log sin CST 

 therefore, d log sin hel. lat. = d log sin CST = d CST cot CST 

 But, log sin SCT = log sin STC + log ST— log r 

 d log sin SCT = — d log r 



or d SCT = - - tan SCT d log r _ _ ^ ^^^ g^T 

 sin 1" 

 therefore d CST ^ — d SCT = x tan SCT 



Hence d hel. lat. (t) = .r tan SCT cot CST tan ^ . . . . (c) 

 d hel. long. (j8) = d TSP 

 log cos TSP = log cos CST— log cos -x 



hence d TSP = cot TSP tan CST d CST— cot TSP tan «-dx 

 or substituting from equation (c) and the equation preceding (c) 

 dhel.long. (iS)=cot TSP tan SCT x (tan CST— cot CST tanV)....(e) 

 log sin A = log cos \ (/S' — /S) + | log cos w + i log cos i/ 

 Hence 2dA=— tan A(d/3'— d|3)tanK/3'— /S)+d^tan ^ + d^r' tan^ (f) 

 and dV=— i (d^ + d^r'+SdA) tan ^ V tan (i t + ^x' + A)) . x 

 — i (dT + d,/— 2dA) tan f V tan (f ^+ ^ — A) ) 



Thus by equation (a) we obtain the value of d/— di' = dU and by 

 equations (a), (6) (c) (e), (f) and (g-), the value of dV and thence 

 the equation U + dU=V+dV in terms containing only known ^^quan" 

 tities and the two unknown quantities dp and At. And in a similar 

 manner the equation 



U' + dU' == V+ dV in terms containing only known quantities and 

 the two unknown quantities dp and d/. 



The solution of these equations furnishes the values of dp and At. 



VOL. XIII. D D 



