of Polarization Plane in an Aeolotropic Medium, 225 



point, &c. — of the Fresnel surface, and no more, are repeated 

 in the electromagnetic surface. 



I am not familiar with the details of experimental work in 

 Physical Optics, and therefore cannot say wmether there is 

 conclusive evidence that the Fresnel surface is accurately 

 the form of the true optical wave-surface. We see by the 

 above statement that the mere qualities of double refraction, 

 conical refraction, &c. will not. serve to distinguish between 

 the Fresnel and general electromagnetic wave-surfaces. 



The notation and language of Quaternions will be used 

 below, and Mr. Heaviside's valuable practice of denoting 

 (whenever desirable) vectors by Clarendon type will be 

 followed. 



Prof. Hatha way's (' Primer of Quaternions,' Macmillan, 

 N.Y., 1896) term " nonion " for "linear vector function of a 

 vector " will be used. [But I should .like, in passing, to say 

 that I think the term a bad one. Some single term is almost 

 indispensable, and I had suggested " Hamiltonian/' Prof. 

 Hathaway rightly objected that Hamilton's name should not 

 thus be appropriated to a minor function occurring in 

 Quaternions. I therefore prefer " nonion.'" But there is 

 this serious objection to thus indiscriminately extending 

 the principle which underlies the formation of the word 

 " quaternion " — that there will be many kinds of qua- 

 ternions, many kinds of nonions, &c. For instance, a unit 

 rotor would be a quaternion, and what I have in Oetonions 

 called a self-conjugate pencil function would be a nonion.] 



If % be a nonion, and y^ its conjugate, the strain corre- 

 sponding to ^~ l may be called the reciprocal conjugate of 

 the strain corresponding to %. It is physically described as 

 follows : — If a given strain be effected by first making a 

 pure strain and- then a rotation, the pure strain may be 

 called the pure part of, and the rotation the rotation of, 

 the given strain. The reciprocal of a given pure strain is 

 naturally defined as the pure strain whose axes are those of 

 the given strain and whose elongations are the reciprocals of 

 those of the given strain. The reciprocal conjugate of a 

 given strain is then one whose pure part is the reciprocal 

 of the pure part of the given strain and whose rotation 

 is the same as that of the given strain. 



We require the following 



Lemma. — If Hue a given surface, and S its polar reciprocal 

 ivith regard to a given origin; and if by a homogeneous strain 

 which leaves the origin unmoved II become 11', and by the 

 reciprocal conjugate strain S become S' ; then S' is the polar 

 reciprocal ofH' with regard to the same origin. 



Phil. Mag. S. 5. Yol. 4=2. No. 256. Sept. 1896. S 



