228 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



is probable that the resulting equation will be of a higher degree than really belongs 

 to the problem in its direct form. If so, it will contain foreign factors, which it may- 

 be difficult to detect and peculiarize, so as to separate them from the proper solutions 

 ol the problem. The method, besides, is not essentially different from the last. 



Another difficulty also presents itself here ; nor is it the only one. The mere in- 

 tersection of the magnetic axis with the magnetic needle is not a test of the duality 

 in point of number, nor of the equality of intensity in point of force, nor is it confined 

 to any law of variation of force whatever ; and hence the mere intersection is not of 

 itself sufficient for the determination of the question respecting the duality or the 

 relative intensity of the poles. Still it is one of the necessary conditions, though not 

 the only one, by which the hypothesis is to be tested ; since the poles, being of any 

 number, and of any intensities whatever, if situated in the same straight line, will cause 

 the needle to intersect that line, and hence render that phenomenon incapable of de- 

 termining the number, intensity, or position of the poles ; yet wherever this intersec- 

 tion is not fulfilled the duality of the poles cannot be admitted, nor yet the position 

 of the poles, however many they may be, be in one straight line. The determination 

 of a^ b, Cj ttii b„ Cji from the equations (20 — 24.) cannot then be effected completely. 



We shall hence proceed in the following manner. A straight line, which constantly 

 touches three given straight lines, but undergoing all the changes compatible with 

 that triple contact, describes the hyperboloid of one sheet. This surface being of the 

 second order, will be cut by a fourth given line in two points ; and hence there are 

 two positions which a line resting upon four other lines can take. If, then, we ima- 

 gine these four lines to be four different positions of the magnetic needle, and the 

 line which rests upon them to be the magnetic axis, we shall perceive at once that in 

 case of any number of poles of any variety of intensity, and acting under any law of 

 variation of force depending upon distance, the magnetic axis can be determined in 

 position from four observations of the magnetic needle ; and, therefore, of course, in 

 the case which we are examining, where the poles are two, the intensities equal, and 

 the law of force that determined by Michel and Coulomb. 



Let us take as the equations of the magnetic axis and four of the needles, respec- 

 tively, the following : 



0? = a 2 -f- a andy = b 2 + |8 



X =z a' « + a' andi/ = b' z -\- (i' 



x = a" z-\-u". eiudy=:b" z + ^" ^ (25—29.) 



X = a'" z + a'" andy = b'" z + ^'" 



X =z a^" z -\- co" and y ^h" z -{■ ^^ . 



Then the condition, that the first of these intersects each of the others simultaneously, 

 gives the four equations, 



(a'-a)(6'-F) = ((3'-^)(a'-a) . (30.) 



