242 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



generate as many sheets of surface as there are branches in the curve ; but as the 

 symmetrical branches of the curve generate superposed sheets of surface, or sheets 

 which are geometrically identical, the number actually described is only half as great 

 as the number of branches. There may hence be one, two, three, or four sheets of 

 surface generated, according as one, two, three, or four of the values oi'if- are real and 

 positive ; and upon any point in one of these a minute needle being placed, and a 

 circle described through that point and the points Q R, the needle will find its re- 

 pose in that plane, and be a tangent to the circle at that point. 



6. The intersection of these four sheets of surface with the earth's surface would 

 give four separate and continuous lines upon the surface of our globe, upon any point 

 of which the needle being placed, it would be horizontal. In other words, there may 

 be four such lines as that which has been denominated the magnetic equator. 



7. Observation, however, seems to indicate only one single branch of this inter- 

 section ; though it must be confessed that the greater part of the observations, and 

 the mode of determination of the position of the equator, are far from satisfactory, 

 llie great difficulty of procuring good instruments, and the almost equally great 

 difficulty in making a correct observation with any instrument whatever, at places 

 which would give results free from suspicion of foreign and local sources of error ; 

 the extremely small number of observations actually attempted, and the very hypo- 

 thetical character of the formula by which the equator is determined from observations 

 made on either side at the distance of a few degrees ; all these reasons, and others, 

 render the delineation of this line, as laid down by M. Morlet, very far from satis- 

 factory. The four branches may indeed be easily conceived to lie so near to one an- 

 other, that of points which have been actually observed or inferred from observation 

 and theoretical reductions, some might be in one and others in other branches of the 

 fourfold system of lines ; and hence that the spherical polygon traced through these 

 might not be in reality composed of chords of any one single branch of the system. 



8. It therefore becomes necessary to examine the curve more minutely as to the 

 number and circumstances of the branches of which it is actually composed on any 

 hypothesis which is consistent with the other phenomena to be accounted for. Since, 

 however, the general equation in terms of x and y is altogether unfitted for our pur- 

 pose, and the equation between the radius vector and polar angle offers no simplifi- 

 cation in the form of the expression, I was led to attempt it by examining the relation 

 which subsists between the angles made by radiants from the poles to points in the 

 curve, with the line joining the poles. This also proved almost equally useless in 

 respect to the purpose I had in view ; but upon trying the radiants themselves the 

 object was completely attained. 



Resuming equation (59.), and recollecting that 



y^ + (« + «)^ = rf 

 y2 + (^ - of- = r/, 



