16 
Proceedings of the Royal Society of Edinburgh. [Sess- 
plane to a calainoid, the conditions that the calamoid is half-parallel and 
half-perpendicular to the electropotential surface become (by (10) and (15)) 
^ "I” ”1“ dyf^ — 0 
4“ dy€. d^Q = 0, 
so the calamoid must satisfy the two equations 
a(rr, y ) _ a(a?^^) 9(a^, t) . S(y, z) 0_(?/,J) t) _ ' 
^d{u, v) '^d(u, v) ' ^d(u,v) ^d(u, v) '^d{u, v) ^d{u, v) 
^ y),^ .7, 4^7 i) 
%u, v)'^ ^d{u, vy ^d{u, v) 
j 5(//, g) . 0( y, 0 , a(y^ 
“0(^^ i.)^ ^0(^z, "0(^^, ?;) 
(16) 
If in particular we take u, to be two of the co-ordinates, say x and y, 
we obtain 
02 
dt 
02 
dt 
/|.02 0i^ 02 0A 
0.r ^ \0;r 0// dy dxj 
, 1 
02 02^ 02 02^ . 
0^c dy dy dx. 
(IP 
These two equations are the partial differential equations whose solutions 
are represented hy the calamoids. 
§ 11. The Integral-Equivalent of Maxwell’s Equations. 
In 1898 M. de Franchis* showed that the theorems regarding the 
equivalence of surface-integrals and volume-integrals in three-dimensional 
space, which are known as “ Green’s theorem ” and “ Stokes’ theorem,” 
can be extended to space of any number of dimensions. If (hx, hy, hz, 
dx, dy, dz) are any six functions of position in our four-dimensional hyper- 
space, one of de Franchis’ formulae may be written 
i f f,, y) ft 
JJ l ^d(u, v) ’'0(w, v) “’d(u, v) 
0 , 7. j ^(v, t) , j 3(2. 0 ‘ 
^ ^Z(u, vy '“'d(u;v) V). 
du dv 
ddz 
dy 
ddy dhyd(y, z, t) ffdz dd^ dhy\d(x, z, t) 
dz dt Jd(p, q, r) 
4- 
dt Jd(p, q, r) 
dhz hd{x, y, z) 
\dx dy ' dt Jd(p, q, r) ' \dx ‘ dy ' dz)d{p,q,r) 
(ddy _ ^ dhz \ d{x, y, t) 
dx 
dh. 
dz 
dhy 
+ U+ 
dp dq dr. 
Here the integral on the left is taken over any closed surface S, and 
the integral on the right is taken over any open hypersurface V which 
is bounded by S. For convenience, position on S is specified by two 
variables (^6, t?), and position on the hypersurface V is specified by three 
variables (p,q,r)] but it is evident that the theorem is really independent 
of the choice of these variables. 
* Palermo Rend., xii (1898) p. 163. 
