OP LUMINOUS VIBRATIONS. .589 



if u = X + y v/ - ! and v = w -y\/ - \, by integrating as before, 



f=F(n) + G{v)-e\vF,{u) + uG,{v)\ +~ \i<' F,{u) + u'G,{v)} - &c (17). 



The impossible quantities are got rid of by making G the same function as F. If then, to 

 obtain an exact value of/, we suppose that F {u) = Ae*" and G(«) = ^6*'', we shall have. 



« 1 . 2 . «- ' 



= Ae (I - — + — — + &c.) + Ae" (1 h + Sic \ 



k l.2.k' 1.2.3.A;= '' ^ A; 1.2.;r 1 . 2 . 3 . Ar' '^ ^ 





= 2.^6^ *•'' COS I A; + - J «/. 



/= 2 4'6^ "■' .cos (A: + T7] 



Since, from the form of equation (12) .r and y are interchangeable, we shall also have 



Therefore generally, 



/= 2^6^'''''^''cos (k + -^ ) y + 2 A' J-' '''^ "cos (k' + ^-\ .I'. 



As the quantities k and k' may be any whatever, this solution is so far indeterminate. But it 

 is clear that the value of / must not, from the nature of the question, increase indefinitely 

 with J.- and y, and that consequently the exponentials must be made to disappear. Hence we 

 shall have k = k' = \/e, and 



/= 2 j^ cos 2 x/ey + 2^' cos 2 \/e,r (18). 



This then is the general form of / expressed in finite terms, and subject to the limitation of 

 being free from exponentials. Other forms may be adduced, apparently, but not really, different 

 from this, which equally satisfy the equation (12). For instance /= ,/ cosi/.i' cos r///, provided 

 ij' + q' = Ve. But this is reducible to the form of the terms of equation (IS), by u change in the 

 direction of the axes of x and y. (See Theory of the Polarization of Light, p. ST.i.) 



I shall have occasion hereafter to advert to equation (iS). At present I have only to remark 

 that the above form of / does not correctly define tlie motion transverse to the axis of ;?, at least 

 for all values of x and y, for this reason. At the boundary beyond which the motion does 

 not extend in directions transverse to z there must be neitlicr condensation nor variation of 



condensation, otherwise there will be transverse propagation. Hence/, — , and - — must 



vanish together. But ])lainly this is not the case with the value of / ol)taiiied above. 



9. From the above reasoning we may conclude tii.it thc> form of / we are seeking for, i.s 

 not expressible in finite terms, and niii-it consequently In- "blained in an infinite series. The 

 Vol.. VIII. I'Ain V. 4G 



