Mr Hargreaves, Examples illustrating the use, etc. 171 
Haamples illustrating the use of Integral forms. By R. 
HarcGrReaves, M.A., St John’s College. 
[Received 16 July 1915.] 
The following are examples of the methods described in a 
paper on “Integral forms and their connexion with Physical 
Equations*.” ‘The first deals with the invariance of the electro- 
magnetic equations under a transformation of time and one 
coordinate without assumption as to the functional character of the 
relations. The second introduces a characteristic or generating 
function for electromagnetic action by analogy with Clebsch’s 
theorem in Hydrodynamics. The third connects Clebsch’s form 
in Hydrodynamics with the methods of the paper. 
ExamPLe I. The integral forms to which the electromagnetic 
equations are related, (18) and (21) op. ctt., are 
QO, (e) = [Xdyde +Vdzd« + Zdardy + Vadtd« + Vbdtdy + Vedtdz 
Ratchaneotes (la), 
and 
0,(m)= jadydz + bdzda + cdady — VX dtdx — V Ydtdy — VZdtdz 
eae (1b). 
Any transformation which leaves the electromagnetic equations 
unaltered in form, must also make these forms invariant. We 
suppose a=’, y=y’, but ¢ and z functions of ¢’ and 2’; and make 
no assumption as to the nature of the functions. Then 0, (e) 
becomes 
Oz ot j 2 Oe ot 27).0 Nap) 
[(xB-"e ss) ay ae +(7 wt Vax) de'da’ + Zda'dy 
ae Wee ea: Ob, UX oz AN 
4V (a5, + ap) lide +V (bo 7 a) dtdy 
(Gb 02 CP G2 saa, 
ae G ae 
and we infer that X’...c’ must be given by 
: Oz ot 7 we OZ ot jae 
A= X VOR, Ne Nie VO ar Ge, 
naa YI yp Xe yg (08H 02) 
mci Vel ae. Vat Ga dz’ Ot! 
* Trans. Camb. Phil. Soc. Vol. xxt. iii. 1908. 
VOL. XVIII. PT. IV. 12 
