REPRESENTING MEASURED ARCS OF THE MERIDIAN. 395 

 1 3G00 , , 



a = - . — — (5-' - s), 

 § 9\ 



b = —< 2«, sm/cos2L— ( — «i2-f — «,Asin2/cos4L V 



we have 



of — X =i in + ai + b k, 



and a similar equation for the combination of the southernmost 

 point of an arc with each of the points to the north of it. 



The sum of the squares of the alterations to be applied to all 

 the latitudes of an arc is thus : 



x^ +{m + ai + bk + xf + (m' -\-a' i + b' k4- xf, &c. ; 

 for other arcs the sums are 



x^ + (mj + c, i + ^1 ^ + ^j)^ + (?«', -\- a\i + b\ k + x-^"^, &c., 



x.^ + (?re.2 + Og ^ + ^2 ^ + '^'2)^ + ^'2 + ^'2 ^ + ^'2 ^ + *'2)^ ^^-y 

 &.C., &c., &c., 



each of these gives thus for the determination of its own value 

 of X the equation 



= /x ^ + (m) + (a) i + [b) k, 



in which jtc is the number of the observed latitudes, and (m) («) 

 and {b) denote as in the usual notation of Gauss. It furnishes 

 also towards the determination of i and k, which must be found- 

 ed on all the existing measured arcs, the following contributions : 



{a m) + {a) x+ (a^) i+ {a b) k, 

 {bm) + {b) X + {ab) i + (b^) k, 

 which, eliminating x, become 



ia «) - Wi^' + {(„=) - WJfi) j i + ^(„ 4) - (AMj,, 



the sums of the first as well as of the second of these contribu- 

 tions, so furnished by all the existing measured arcs, being made 

 = 0, give the two equations necessary for the determination of 

 i and k. 



§.3. 



I will now communicate the several equations of condition 

 which I have deduced from each of the ten arcs on which 

 this examination is founded. My view in so doing is to obtain 

 the advantage of being able to avail myself of any subsequent 



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