Prof. Challis on the Undulatory Theory of Light. 333 



fact, from the foregoing values of w and a>, it will be seen 



that the ratio of -^7- to kv is that of e to — . Hence if 8h=$l, 

 oh ol A 



we have ^— = — , which is clearly the ratio in which the two 



velocities contribute to the changes of condensation. 

 1 ~\ 2 



By substituting — r for — ~ in the value of k, we obtain for the 



velocity of propagation a\f 1 H — f \ which shows that the 



excess of the velocity above the value a is caused by the trans- 

 verse velocity, and that, because the changes of condensation are 

 due to the transverse as well as the longitudinal velocity, they 

 are more rapid than they would be if due to the latter alone, and 

 the rate of propagation of a given state of density is consequently 

 accelerated. 



Reverting now to the expressions for a>' and 0-', if (2?z + l)7r 



be substituted for c 7 , rr for 2 */e 3 and k' for -— , the following 

 equations may be obtained : — 



a>'= — -f-^cos— -( sm-T-y (h — tcat)+ srn-y ("+*^ B }> 



, 27rmfc'f () 2ttI ( . 2ir tl , . . 2ir ,, , .A 

 & — — , -cos -^— I sin -7- (h— fc'at) — sm ^-j (h -f tc'at) I . 



A CI At V Ai A / 



At the same time 



, 27rm/ n 2irh ( . 2tt fl jN . 2tt /7 jX \ 

 m/ = — _£it C os-— r 1 sin — - (l—jcat) -\- sin — (l+fcat) j 



A A \ A A / 



2irmKf 27rh / . 2tt ,, . . 27r 



27rA / . 2-7T , . . 27T /7 N \ 



—j-— 7— I sin— (t — /car) — sm — (t-f #ar) j. 

 Aa A \ A A / 



Hence the transverse motion and condensation may be repre- 

 sented by equations exactly analogous to those which represent 

 the longitudinal motion and condensation, and the two motions 



are correlative to each other. If .-^ be substituted for e in the 



value of/e,we obtain k — \f 1 + -j- 2 . Also /c l = — =-\f 1+— %. 



Thus the velocities of propagation kci and da are each greater 

 than a, because, as already explained, the transverse and longi- 

 tudinal motions both contribute to the changes of condensation. 

 But the ratio of these velocities is that of a to a', as evidently 

 should be the case, since the propagations over these breadths 

 occupy necessarily the same time. 



