Action of a Galvanic Coil on an external small Magnet. 439 



It is here to be remarked that in Ampere's theory the proof of the 

 law of the inverse square is founded upon the experimental fact ; 

 whereas in the course of reasoning which I have adopted that law 

 is deduced from independent apriori principles (see art. 57) ; and 

 the above explanation of the fact may therefore be regarded as 

 evidence of the truth of the hydrodynamical theory of the gal- 

 vanic current. It may also be remarked that the above result 

 is the same as that obtained by Ampere, although I have taken 

 no account of the term in his formula which involves the factor 

 Jc. The fact is, as the analysis in Jamin pp. 205-207 shows, the 

 result is independent of the particular value of k, and would be 

 the same if this factor were supposed to be zero. 



70. I propose now to calculate the action of a finite recti- 

 linear rheophore AB on another CD, also finite and rectilinear, 

 supposing their axes to be in the same plane and inclined to each 

 other at any angle, the latter to be fixed and the former to be 

 susceptible of motion in that plane about a centre situated 

 at the intersection of the prolongation of the axes (seethe figure 

 555 in Jamin, p. 200). The extremities C and B being those 

 nearest to O, suppose the direction of the current in CD to be 

 from C towards D, and that of the current in AB from A towards 

 B; and let OC = «, OD = 6, OB = a', OA = 6', and the angle 

 AOD = w. Also, P being the position of any element of the 

 fixed rheophore, and P' that of an element of the movable one, 

 let OP = 5, OP' =s', the angle OPP' = 0, and the angle OFP = 0'. 

 Then, F being the action of the element at P on that at P', the 

 formula (c) becomes for this case 



r 2 



As the movable rheophore is restricted to motion of rotation 

 about 0, in calculating the force tending to put it in motion we 

 have only to take account of the resolved part perpendicular 

 to its axis, which is F sin 6'. Hence., in order to calculate 

 the sum of the moments of the forces about 0, it will be re- 

 quired to find the value of X . F s' sin 6 1 , or of the double 

 integral 



sin 6 sin 2 & ds ds\ 



the first integral being taken with respect to s, from s = a to 



s=b, and the other being taken with respect to s ! , from s' = a l 



to s'=Z>'. For effecting these integrations we have the three 



equations 



s_sin0' r _sin o> 2 8 f9 . . 

 -7=^ — n> -i =- — nt r z = s z -{-s^ — 2ss' cos co. 

 s 1 sin s' sin 6 



