CONCERNING TERRESTRIAL MAGNETISM. 227 



that the problem is now reduced to a purely arithmetical state, requiring only the 

 application of known processes, and perfectly capable of execution, for the actual 

 assignment of the positions of the poles themselves. A very slight attention to the 

 processes themselves must, however, convince us that the operations will be very 

 laborious ; but at the same time, the symmetrical forms in which the two triads of 

 equations are presented, might induce us to hope that a gi-eater degree of simplifi- 

 cation would result in the final formulae than our passage through so many operations 

 could at first sight lead us to expect. Such, indeed, proves to be the case ; and the 

 results are not altogether destitute of elegance, as well as simplification. Fortunately, 

 however, there is no necessity to even attempt the solution under the present aspect 

 of the problem ; and having learnt from it, in its present state, the number of obser- 

 vations necessary for the determination of the poles, we shall exchange it for another, 

 which is in some degree different as to general plan, and considerably more simple 

 in its requisite calculations. 



VI. — A necessary consequence of the hypothesis upon which we are proceeding, 

 is, — that all the needles must intersect the magnetic axis. If, then, we assume the 

 coordinates of the two poles a^ b, c, and a^^ b,, c^^ we have the equation of the magnetic 

 axis as before, 



a„— a, a„c, -^ a,c„ -\ 



(15.) 



X = -r — . z 



I _ "II "I — '*/'"// 



C/i Ci 



And if we take the equations of four magnetic needles, as 



X = a' z -{- (x! and^ = b' z -\- p' ^ 



x = a" z + a" andy = b"z + Q" I 



^ ^ > (16—19.) 



x = a"'z + a"'axidi/=zb"'z + ^" | ^ 



X =z a^"^ z -{- u^ and y =^b^^ z -{- 13^^, J 

 the intersections of (15.) with (16 — 19.) give four equations, of condition similar to 

 those of (12.), (13.), (14.), from which to determine o, b^ c, and a^^ b^, c^^ viz. 



(«' - "-^^') (*' - 1^;) = (^' - ^t^) («' - ^;)- • • (20-24.) 



//» 



These four equations will determine four of the coordinates, as a^ b, and a^, b 

 leaving the two others indeterminate. But still the law of force furnishes four other 

 equations from which to determine the two quantities c, and c^^ ; that is, a redundancy 

 of equations, from which redundancy the remaining number may be taken as checks 

 of accurate calculation if the principle be admitted, or as tests of the truth of the 

 principle when we are assured of the accuracy of calculation and of observation. 



By this method a greater uniformity of process, and a perfect symmetry in respect 

 to the quantities involved, are obtained ; but still the process is very laborious, and it 



2g2 



