OF LUMINOUS VIBRATIONS. 587 



it is clear that G must be a circular function. Let therefore G{v) = Ae'"'\/^^ + Be""''^~; or, 

 what is equivalent, let G (v) = mcos(7iv + c). Then, 



^ , ^ r I , , "^ ■ , ^ m d . cos (nv + c) 



G, (V) = /mcos(rau + c) dv = — sin (nv + c) = . ^ 



J n n^ ills ' 



^ , ^ r"^ ■ f > ■ '" , »> 



Gi (o) = /— sin (nv + c) dv = cos (nv + c) = — 



' J n ' n' ' n" 



ri^ dv 



m d? . cos {nv + c) 



; cos (nv +c)dv = sin (nv + c) = 



n- 71^ ' n' 



i 

 Consequently, 



dv' 

 m d' . cos (nv + c) 



dv^ 

 &c. = Sic. 



m d.cos(nv + c) m d'.cos(nv+c) e'u^ 



m — m cos (Ml) + c) . ; . eu ^ . ' . 



w dv «* dv^ 1 . 2 



m d^.cos(M» + c) e^v? 



«' dv^ 1 . 2 . M 



+ &c. 



= mcos \n\v -\ + c\ 



= mcos \7i(z — at) (z + at) + c} 



= m cos {w )^~ \n ^ — \ at + c\. 



w + - = V -rr + *^- 



Let, now, w = — . Then n + - = \/ — ^ + 4e. We have, therefore, finally, 



n \ ,, \' 



oj; v 1 + -^ + c) = \// 



I 



/ ^ = m/ cos — - (jr - o)"V 1 + -^ + c') = ^/'■ 



A. TT 



The velocity in the direction of ^s: is / ■— . Hence, if w^ = 



= m,/sin — (jr - o< V 1 + "t +'') 



.(15). 



Also, since (Art. 3) /.-p-+ a' « = 0, it follows that 



« = -- - 1: = ra,/ V 1 + —J- s'n— (z - at \/ I + -- + c) (16). 



a dt 



/ eX* 

 It hence appears that the velocity of propagation of the wave whose breadth is X, is a \/ 1 + — -j . 



The value of e depends on equation (12). If the velocity of propagation be independent of X, 



L ,1 L eX'' , . , , k'K- 



we shall have = k, a. nunicncal constant, and consequently e = -—^ . 



tt' X 



5. Since equation (g) is linear with constant coefficients, it will be satisfied by the sum 

 of any number of such values of \1/ as that just obtained, /, c, m, X, and c', being different 

 for each. Hence we have generally, 



