236 Prof. A. McAulay on the 



and re-writing (4) to (8) 



8 = Yya, /3 = Yae, (12) 



S8 e =S£ 7 ==-i, (13) 



<7 = VS/3, (14) 



0=SSi/€ = S€r/S = S/8c?7 = Syrfft . . . . (15) 

 P = Yey (16) 



From (2) and (11) 



8 = ce, f3=fiy (17) 



Hence from (13), 



Sece=— 1, S7/*7=— 1, ") Mft . 



SSc- 1 8=-l, S^-^=-l.j- vl») 



This is Prop. I. Prop. II. is contained in equations (9), 

 (10), (13), (15). Prop. III. is contained in equations (12), 

 (14), (16), and 



e=V/3p, y=Y P B, (19) 



which are simple deductions from (16), (12), (13). 



To prove Prop. IV. notice that equations (18) are the 

 equations of the E-oid, H-oid, D-oid, and B-oid respectively. 

 From the equations 



S0e=O, SS 7 = 



[Prop. III. or eq. (12)] we have by (17) 



S e///y = 0, $yce = 0, 



which express that e and 7 are conjugate to one another with 

 regard to the H-oid and with regard to the E-oid [these being 

 Syfjb<y= — 1 , Sece =■ — 1 , respectively] . Similarly for 8 and ft. 



All these geometrical properties have been proved without 

 finding the equations of the wave-surface and index-surface, 

 and the other well-known properties of the wave-surface can 

 be similarly proved. But of course for some purposes it 

 may be desirable to find these equations. 



The finding of the equation of the index-surface (by 

 essentially the same method as Mr. Heavi side's) may be 

 thus put 



C€ = S=Y7o-= V(fju- l p<r) = -V<r/ir l Y&e 



= ^m~ 1 /JbYfji(rY<T6, 



where m is the product of the three principal permeabilities. 

 Thus 



