234 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



XII. — We may now proceed to compute the equations of that straight line (or lines, 

 for there are two, but determined by the same series of processes,) which rests upon 

 any four of lines determined as those in the last section were found. Thus we shall 

 take as an instance Chamisso, Valparaiso, Paramatta, and Port Bowen, the equations 

 of which are given in the Table. 



Assume as the equations of the magnetic axis the two following : 



X =: az-\- a, 



and denote the several equations of the needles by equations of the same form, but 

 with the constants accented, viz. a' b', oc' |3' : then the conditions of intersection will, 

 in each case, be expressed by the equation 



(«' - a) (b' -b)- (,G' - (B) {a' - a) r= 0. 



The insertions of the actual values in the four cases mentioned above being made, 

 and the vincula thrown open, we obtain 



•055920 + -690471 a — '081565 (5 + '248470 a + '051635 b + ub — a(i = 0, 



— •911471 — -621395 05 —3'1 59630 f3 + '206769 a +2^518180 ^> + a Z> — rtj8 = 0, 

 •473095 —7-349865 a —3-418875 |3 +3-363313 a -\-l'500l20 b -\- ub — a|3 = 0, 



— •008188— •034611a- •278746(3- -027756 a + ^023702 6 + afe — a(3 = 0. 



Subtracting each of the last three of these from the first, we obtain 



b = -391938 + •531872 a +1-247954/3 + -017024 a, 

 b= - -288008 + 3-550860 a +2-304000 13 - 2^150420 «, 

 b = —2^295058 -25-957900 a —7*059053 13 —9*888800 a. 



Subtract the two latter from the former of these, then there result 



«= —-271254 — 2-674164a — -838603/3, and 

 a = — -313706 +1-392881 a +-487231 (3. 



From equating which values of a we obtain (3 = -032019 —3-067660 a. 

 Insert this value of j8 in that of a, and we find a = — -298105 — '101755 a. 

 In a similar manner, from the proper substitutions, 6 = '426822 —3-288083 a. 

 And inserting all these values in the first of the four equations, that of the Cha 

 misso needle, we obtain 



a2 —'071009a = '003821, and hence a = '035505 +'071274. 



The two pairs of equations which result from this calculation, then, as those of the 

 magnetic axis, are 



x=: --308970 5J +-106779r, 

 1^ = -075732 2 —•295538 r ; 



and X = —•294465 z —•035769 r, 

 y— •544430 5; +^141744r; 



according as the + or — sign is taken in the valuation of a. 



In the same way we may proceed to find the magnetic axis which would accord 

 with any other four observations, and by a comparison of these ascertain whether the 



