EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
18 
according to the rules of the Calculus of Variations. If then, in the variation 
2 ^dt + vhv + v'hw — { fSf + gSg + hSh)]dT, 
we replace Sv, . . . l)y a second system"^ of possilde displacements k', . . . we shall 
obtain a symmetrical expression 
+ i'v + — C' {ff' + gg + hh')]dT . (20), 
wliich admits of a similar transformation ; and the result ol>tained, wlien simplified 
h}" means of the equations of motion, will consist of the volume integral of a ])erfect 
difterential coefficient with respect to f, and a surface integral. The symmetry of 
tlie expression (29) tlien leads to the reciprocal theorem. 
We have 
{uu + vv + ini') dr = [|f — {iiu' + vd + wii/)dT, 
the volume integrations being taken through the space bounded by the surfaces S. 
Also, denoting by /, m, n the direction cosines of the normal to S drawn into this 
space, we have 
iff' + ffl/ + dr = — II {{g7} — hm) u + {Id — fn) d + (fm — gJ) a/jdS 
% / /T// 3/'\ , /3y’ dA /] 
0 // 
a-' ar; 
-d:.)n-\-[^-i]d+U-^]w\dr. 
di/ a./ 
Tdence the expression (29) becomes 
''0//. 0// 
0 / 0 /( 
0 [dj oh 
(_f f [n'|« + - abl +»’ f ‘ - g j} + + o= ( y - /j j 
dr 
+ I j j ^ {nu' + i'd + nud) dr + 0^ | { {gn — hm) u' + (/d — fn) d + (/in — gl) w'] d8. 
The first line vanishes identically; and, from the symmetry of the expression (29), 
we deduce the reciprocal theorem 
|j + dw')dT + {{gn — hrn)v' + {Id — fn) d + { fa — gl)w'] dS 
ijt aat)(lT + c° JJ {(</'» — h'm)u + (h'l —f'n)v-\-{/'m — fj'l)w] c/S 
. . . ( 30 ). 
* The second system, as well as the first, satisfies the fundamental equations. 
