230 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



VII. — Having obtained the equations of the magnetic axis by the determination of 

 ab a ^, we may put the rectilinearity of the positions of the poles to the test at once. 

 For if we take a fifth needle 



x = a"" z -{- ct' and 1/ = b"" z -\- P'' (34.) 



and combine it with (29.), viz. 



:t; = a z -\- oc, and 1/ =: b z -\- fi, 

 so as to ascertain whether they intersect or not, by means of the equation of condition 



(a^ -'^) (J^ -1) = (j3^ -"^) (a^ - «) ...... (35.) 



The approximation to a fulfilment of this condition for a^ a"", i^ j8% derived from all the 

 other observations upon which reliance can be placed, and to the extent of the probable 

 accuracy of those observations, will establish this hypothesis to the same extent. 



In order to estimate the real amount of the error in the application of this equation 

 of condition, we must recollect that the formula itself is derived from that which 

 gives the shortest distance between two given lines, viz. 



D (dist.) = + (-: - "-^ ^^' - ^) - ^^' - ^^ ^^' - ^^ (36.) 



y (flv _ af + (6^ — bf + (a 6^ - a- bf 



Hence, in order to estimate the number of miles which the needle would be from 

 fulfilling the condition, we must calculate the denominator of the fraction (36.), and 

 divide the result of (35.) by it. By then calculating the angle which D would sub- 

 tend at the place of observation, we shall be in some degree prepared to judge 

 whether such an error might possibly have arisen from unskilful observation, the im- 

 perfect structure of the instrument, from any probable geological or meteorological 

 causes, or from any temporary local disturbance. Or, conversely, were this line satis- 

 factorily determined by a great number of tests, and the instrument and observer 

 well prepared for the task, then we should be able in some degree to estimate the 

 amount of the disturbing forces that climate, geological structure, and local attrac- 

 tion do actually exert at that place, and perhaps in some cases also to form a pro- 

 bable conjecture respecting the separate contribution of each of these causes to the 

 total amount of the disturbance. 



VIII. — Let us now suppose the magnetic axis satisfactorily determined and tested ; 

 and proceed to inquire whether the poles be two or more, and whether equal or un- 

 equal in the intensity of their action : and in the first place we shall suppose the 

 intensities equal. 



Recurring to the figure in (III.) and the conditions that are tabulated, as (5.), (6.), 

 (7.), we have the quantities designated as a, b, c, A, B, C,f, g, h, and the correspond- 

 ing ones for any number of points M^ N^ P^ Q^, &c. from actual observation and the 

 calculated equation of the magnetic axis. From any two of these, as (5.), (6.), we 

 then shall be able to compute t and u. The remaining equations, whatever be their 



