PAPER BY PROF. HELMHOLTZ. 107 



The quantity z determines the altitude of the wave, which according 

 to equation (la) is — 



* = 2£.-lognat.(j±£). 



The three equations referred to may now be written : 



I. z {£ [2 - 2z 2 + i;sj + sp [2 + fC 2 ] - (1 - C 2 )} = 0. 

 II. D [2.c 2 — l' 2 ] - sp . c 2 _ ^ + i^ = o. 



III. ; D [2* 2 - f ;«] + $ . I - ie 2 + iC 2 } = 0. 



Of the four quantities that occur herein any three may therefore in 

 general be determined by the fourth. Only one system of values, 

 namely, 



z = and Q + $ = £, 



leaves £ undetermined. This solution holds good for the entire lower 

 wave, for which z is to be neglected as compared with Q. 



Since in general one of the four quantities in the equations I to III 

 remains undetermined, therefore for given properties of the medium 

 and for a given strength of the wind, there remains always one varia- 

 ble parameter of the stationary wave; and in fact the further investi- 

 gation shows that this variable is connected with the quantity of energy 

 that is accumulated in the wave. 



The simplest method of computation is to express the remaining 

 quantities as functions of the cos £. 



~ _ 7 COS 2 £-— fV 



36* cos 2 £-§ 



m „ -v 1 (COS 2 £ — £).(COS 2 £ — #) 



sp = -icos 2 f + ££ = -^- -^e-f 



C 2 [£(cos 2 £-!)-n< + £] = Q + *-£ 



Since D and ^ must necessarily be positive, it follows from the first 

 of these equations that 



or, 



cos 2 £> § = 0.666667; 



cos 2 £<-& = 0.642857. 

 The equation for ^ would also allow cos 2 £>§, but 



0.5 < cos 2 £< 0.642857. 

 Finally the equation for C 2 can be written 



(0.68615-cos 2 e) (cos 2 £+2.18615 ) 

 l _u.4 x (cos , £ _ 0>G65 37) (cos 2 £+1.46537)' 



