382 M. Ampere's Memoir 



which we must take between the limits /3' and /3". We have 

 also the difference of the two functions of the same form, one 

 of |S", the otlier of /3', which must be again integrated, in 

 order to obtain the rotation sought for : it is enough to make 

 this second integration upon one of these two quantities : let 

 a" then be the distance OL" which answers to j3", we shall 

 have 



, o"sin(/3"-£) ,, ... ^ ail ^ I o" sin t d /3" 



y = A- — -= a" cos e — a" sm e cot Q", d s = — : — -;-— : 



and the quantity which we shall wish to integrate first will be 



\a"ii> I 



cos i COS /5" d ^" -,, \ 1 a } 



.n.^.^' + cos(/3"-.)d/3 ;, 



the integral of which taken between the limits /3", and ^"„ is 

 J a'i I J sin (W- .) - sin W-.) _ ^, + ^ | . 



Designating by p,',' and })J, the perpendiculars lowered from 

 the point O on the distances L"L/; = ?;/', L"Ly = ?•/', we have 

 evidently 



a" sin {^J'-,)=pJ', a" sin ^!'=pl', -r^ = ^, -^ = r/L, 



\ru I ruf ri -f^''sin/3,; sins' wn/3/' sm »' 



and the preceding integral becomes 



I ii' IpJ'- Pi' -{rj'-rl') cote-]. 



If we notice that by designating the distance OL' by «', we 

 have also 



, a'sin(/3'— s) , , . ^ />/ j J a'sini 



s' = A-r — = a' cos E — a sm e cot a'as = -.-vtt j 



sin "^ ' sin» IS, ' 



we easily see that the integi'al of the other quantity is formed 

 by that which we have just obtained, changing p^j', p/', r", r" 

 into Pili pI, rj, rj, which gives for the value of the momentum 

 of rotation which is the difference of the two integrals 



I ii' [pj'-pj - P,!+pI - {rJ' - r< - r< + rj.) cot e]. 



That value is reduced to what we have found above, in 

 tlie case where the angle e is right, because then cot £ = 0. 



If we suppose that two currents proceed from point O, and 

 that their lengths OL", OL^^ (fig. 5) are respectively repre- 

 sented by a and ^, the perpendicular O P by j», and the di- 

 stance L"L;, by r, we shall have 



5 ii' [t? + (a + 6 — r) cot e], 



for the value which, in this case, the momentum of rotation 

 takes. 



The quantity « + 6— r, the excess of the sum of two sides of 

 a triangle on the third, is always positive ; whence it follows 

 that the momentum of rotation is greater than the value | i i' p 

 which it takes when the angle e of the two conductors is a 



right 



