Magnetic Force and Torsional Strains. 
17 
Thus, if a and (3 be the y and x components of the strain, then ac 
cording to Saint Venant 
a dy dy 
^- T ° + dx’ a=TX + ~dy- 
If n is rigidity, the forces are n times these; hence + /3 2 ) taken over 
a section of the prism represents the energy of the strain ( potential ) per 
unit of length of the prism, y, which denotes the warping, fulfils 
Laplace’s equation, viz.: 
F V=0, and also-g^r^ 
Here p denotes a perpendicular drawn from the center of the prism (or 
origin of coordinates) to a tangent to the surface of the same. |^For cur¬ 
rent along a cylinder where the current density is F Q and where 12 de¬ 
notes (polar) magnetic potential we have an analogous expression 
dFl „ d{r^ 
■7tr 
’] 
dp ° ds 
It is needless to point out that negative 12 represents y, and TtF Q repre¬ 
sents \r in the elastic analogy. 
In the ordinary notation of strains we have in such prisms under 
torsion P—O, Q=0, R=0, U—O, while 
dy\ 
—TV 4 -— 1 
S '= n ( rx + ^d T=n( r ry+ 
dy/ V y dx/ 
These cover the tractions and shears in the interior, or the body forces; 
while F—O, G—O, FI=T sin cp + S cos cp, reach the surface tractions. 
Consequently 
H=n( dr 
dp 
-rq 
or 
H=n 
i.~dy C ° S Sin sin V~ x cos j 
if p be the above perpendicular and q= distance from point of surface 
for which H is taken along the tangent to the foot of this perpendicu¬ 
lar p. 
For the usual form of couple, according to Coulomb’s law, we should 
have as above 
N—nrj'J(x 2 +y^)dxdy, 
which would require a valne of H equal to that just given to prevent 
warping; while if H be zero and warping freely allowed, the real couple 
necessary to produce the rate of twist (r) would be less than this, i. e., 
N=Jj (Sx — Ty)dxdy=nr j' f'(x 2 + y*)dxdy—nj'f — x ~dJj) dxdy 
3— A. & L. 
