MR. G. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
G 84 
u~ 
Ho 1 - = 
d n , Tr dV 
— + Ku — 
ay dz 
( 30 ), 
H.(i-5r) = 
dSl rr d'i' 
& - k “a7 
( 31 ), 
while the equations satisfied by 'F and fl become 
V? \ 
rP'F 
d 2 ^ d-V 
47T / V? 
+) 
dr 2 
+ 
d f + dz* ~ ' 
■v 1 — — 
K \ v- 
U“ \ 
dm 
dm dm 
+) 
da? 
+ 
df + dz* 0 • 
(32), 
(33). 
The solution of any problem depends upon finding functions which satisfy (32) and 
(33), and which fit in with the particular electromagnetic system which is supposed 
to be moving. In all ordinary parts of the field we shall have p — 0, and thus 
generally we have 
v? \ d 2 'F d'F d*V 
~ v*J dx* + dy* dz* _ 0 
(34). 
Our knowledge of functions which satisfy Laplace’s equation helps us to find 
solutions, for if f(x,y,z ) satisfies — 0, it follows that f{x /\/1 — u*/v 2 , y, z } 
satisfies (34). When in this manner values of T and f! have been found, the values 
of E and H are at once deduced from equations (26) to (31). 
The quantity 1 — id/v* occurs continually in the course of the work, and will 
always be denoted by a. The motion will always be supposed to take place parallel 
to the axis of x, unless it is otherwise stated. 
Application of Vector Methods. 
5. The solution of the six equations typified by (20) and (21) is tedious by ordinary 
algebraical processes. But the solution is readily obtained by simple vector analysis, 
and affords a good example of the great saving of labour effected by Mr. Heaviside’s 
methods. Thus, let F = — V'P and R = — VB, so that, by (18) and (19) 
E-pVHu = F.(35), 
H + KYEu = R . . . ..(36) 
Then we have to find E and H in terms of 'F and B, or in terms of F and R. 
Substituting from (36) in (35) we have 
