644 



Mr. stokes, on THE FORMATION OF THE CENTRAL SPOT 



get two new systems by writing vt + — ior vt. By combining these four systems by addition and 



4 



subtraction, which is allowable on account of the linearity of our equations, we should be able to get 

 rid of the imaginary quantities, and likewise of the exponential e*'"', which does not correspond to 

 the problem, inasmuch as it relates to a disturbance which increases indefinitely in going from the 

 surface of separation into the second medium, and which could only be produced by a disturbing 

 cause existing in the second medium, whereas none such is supposed to exist. 



3. The analytical process will be a good deal simplified by replacing the expressions (A) by 



2ir 

 the following symbolical expressions for the disturbance, where k is put for — , so that kv = k'v' ; 



A 





(B). 



In these expressions, if each exponential of the form e^^ ' be replaced by cos P + v — 1 sin P, the 

 real part of the expressions will agree with {A), and therefore will satisfy the equations of the pro- 

 blem. The coefficients of v — 1 in '^e imaginary part will be derived from the real part by writing 



t -i for t, and therefore will form a system satisfying the same equations, since the form of these 



equations is supposed in no way to depend on the origin of the time ; and since the equations are 

 linear they will be satisfied by the complete expressions {B). 



Suppose now I' to become greater than 1, so that n becomes i/^/— i. Whichever sign we 

 take, the real and imaginary parts of the expressions (B), which must separately satisfy the equations 

 of motion and the equations of condition, will represent two possible systems of waves ; but the 

 upper sign does not correspond to the problem, for the reason already mentioned, so that we must 

 use the lower sign. At the same time that ?i' becomes c'\/- 1, let a, a^, a become 



pe " ', jO,e' ', 

 then we have the symbolical system 



p e ', respectively : 



y _ -e^ -\ _^li{vt-lx-n!)\/ -\ 



V = 



P.' 



-«.\f-l *(ti/-/j- + nz)\/^ 



y _ '-ii\r^_^-hvz^^k\vt-rT)sf^ 



(Q, 



of which the real part 



V = p cos [k(vi — Iw — nz) - 0J "j 

 V^= p,coi\k(vt-lx+nz)-e}, \ ... (Z)). 

 -cos {k'{v't-l'a!)-6'}, \ 



forms the system required. 



As I shall frequently have occasion to allude to a disturbance of the kind expressed by the 

 last of equations {D), it will be convenient to have a name for it, and I shall accordingly call it 

 a superficial undulation. 



4. The interpretation of our results is not yet complete, inasmuch as it remains to consider 

 what is meant by V'. When the vibrations are perpendicular to the plane of incidence there is no 



