258 



Prof. C. Niven on the Theory of an 



These equations will be satisfied by w u -sj- 2 ? OT 3=/> 9> ? l • e> 

 where 



(**, 



f= — — (Ix + my + nz — Yt), 



A. 



and where also 



and 



lf-\-mg-\-nh — 0, 

 V 2 / = a\f- W +fi J^l (mh - ng) " 

 Y*g=b\g-mQ + p*/~\{nf -lh\ 

 Y 2 h=(^7i-ne + fis/~^iQg -mf), 



(23) 



= ^//+^^ + ^/*, 

 f i=^(a 2 l' 2 + b 2 m 2 +...+2f 2 lm)< 



(24) 



Let us now resolve the left-hand members of (23) along 

 (1) the semiaxis Vi -1 , (2) the semiaxis V^ 1 , (3) the normal to 

 the wave ; that is to say, let us choose these lines as the new 

 axes of #!, y x , Z\. The new components of rotation will be 

 (/ 1? ^ 1? o) ? and the direction-cosines of the normal will be (0,0, 1); 

 while the known cogrediency with lines of the terms mh—ng, 

 nf—lh, Ig— mf enables us to write down at once the result of 

 transforming these terms. Coupling this observation with the 

 Lemma expressed in equation (22), the result of the transfor- 

 mation of axes will evidently be as follows : — 





(25) 



These will be satisfied by taking /j = P, g x — — eP\/— 1, the 

 corresponding parts of the rotation being 



P(cos f + V^l sin J), eP(sin ?- V-T cos ?). 



If we consider only the real parts df> & of these we have 



tf=Pcos?, ffi = ePsin£ 



which correspond to an elliptic wave. We may evidently 

 produce this real wave by compounding two imaginary ones ; 

 and we may thus at once proceed without further remark to 

 the discussion of the imaginary wave. Substituting these 



