applied to Electric Currents. 341 
along three axes properly chosen, a, 7’, 2', the nine direction- 
cosines of these axes with their six connecting equations, which 
are equivalent to three independent quantities, and the three 
rotations @,, 0, @, about the axes of 2, y, 2. 
Let the direction-cosines of 2! with respect to 2; y, 2 be 
J}, m,,,, those of y’, 7,, m,n, and those of 2’, U5, mg, ng 3 then 
we find ) 
aS Phd ar the + ie 
dx 
# seam ston sim tte: (64) 
d Sx! Sy! Sx! 
Gq, ot =i =r 3; Pie y! + dstts yp — + gy | 
with similar equ uations for quantities involving dy and dz. 
, Let a', B', y' be the values of a, 6, y referred to the axes of 
2’, y'; a; aa 
a’ =latmB+ny, 
falatmetnn | eRe ee OO 
y! =lsa + ms8 + ngy. 
We shall then have 
da=/ da! + / 08! + [359 + 78, — 88s, ie on (66) 
= hel +18 + hy +10,—80. (67) 
By substituting the values of a’, 6’, y', and comparing with equa. 
tions (64), we find 
d 
d 
ba= a Be BT Baty 7 Be a PL ee (68) 
as the variation of « due to the change of form and position of 
the element. The variations of 8 andy have similar expressions, 
Prop. XI.—To find the electromotive forces ina moving body, 
The variation of the velocity of the vortices in a moving ele- 
ment is due to two causes—the action of the electromotive forces, 
and the change of form and position of theelement. The whole 
variation of @ is therefore 
dQ dR d d 
See (2-5 7, Jt a oe WHOS B04 9-7 8e. « (69) 
But since @ is a function of a, y, 2 and ¢, the variation of # may 
be also written 
Sax Ada 4 o = by 4 moult oat. 
dx 
dz dk be 6 . . e (70) 
