of Electrons in an Elastic Solid JEtlier. 143 



Putting in the last u=u + ^Ve(), v' = u — k Ve( ) the linear 

 terms in e clearly disappear, and the quadratic terms in e 



=isr,r 2 (Ver 1 )(Ve? 2 ) 



= JS6( &SV<,Vef 2 - VefeSfe* + V«&S&»&) 



= 0. 

 Thus we have 



S66Sf,ufe*S66i»{iw6*S66i/6»'6. . (41) 



In the last expression put 

 and apply Th. 4. We get 



= S V1V2TO2 — S Vi VsVa^S^i^ + i S Vi V 2 V3 V^i^S^^. 

 Hence 



/"-=(|SViV 2 V3^Sw3-JSViV 2 V 8 Si ?lM3 ) 



-JS.ViV 2 V3V4S W3 S W4 . (42) 



We now transform 6 ; , s", 5'" so as to be expressed in terms 

 of 



(1) divergence m' and curl e, 



(2) m" and m'" because of Th. 11, 



(3) 8e X e. 



Our choice of these is largely determined by counting 

 the simplest scalar functions of orders first, second, third, 

 that can be formed from them, thus : 



Number of 

 Order. Scalar functions. scalar functions. 



First. m'. 1. 



Second. m' 2 , m' f , e". 3. 



Third. m'% m'm", m'e*, m"', Se % e. 5. 



Now it is easy to transform each of these, and also s\ s", 

 /" (except for the fourth order part of /") into terms of 

 the following: — 



Number of 

 Order. Scalar functions. scalar functions. 



First. SW 1. 



Second. (SV^) 2 , SViV2^ 7 /i ? 72> SVi^7 2 SV 2 ^i. 3. 



Third. (< SV * SV^SViVaS^^SV^SVi^SV^V 



