and the Laws of Plane Waves propagated through them. 



119 



we may immediately deduce, by the formulaj for transformation of the indepen- 

 dent variables, the following values for the nine quantities (a,, a,, &c.), 



a, = a-a'i + i-/32 + c-y'i + hcu' -(- acv' + abw', 



/32 = a'-a\ + h'-^:, + c'% + b'c'u' + a'c'u' + a'b'w', 



73= a"-a; + 6"-|3; + c"-V; + ^'"c"m' + «"c'V + «"i"2o'. 



r a' (a'X + ^""^ + c"aO 



[+c'{a"y\ + b"^^ + c"^',)\ 

 r a" (aa; + 5c4 + raj)-) 



l+c"(aY;+ 67^ + 6-7;)] 



r a (a'a'i + 6'a2 -f c'a'^) 

 a. = ' + 6 (a'/3; + h'a!, + c'/30 



[+c(a'7; + Z''72 + c'70. 



r a" {a'a[ + />'«: + c'a!,)- 



r2=\+b" {a' ^[+b'^', + ,■'§',). 



[^c"{a'y\ + b'y', + ,'y',)_ 



r a {a"a\ + i"a^ + c"aOl 

 a,= \+b{a"^[ + b"^-, + <:"^',)\ 



L+ ^' («"7; + b"y:, + c-'VOj 



r rt' {aa[ + Z*a.^ + caj)-] 

 [+c'(a7; + ft7^ + C7;) 



(19) 



It is plain from the first three equations that the change of the co-ordinate 

 axes will introduce into the function F three new variables {u, v, w). Hence, 

 the function which I have assumed is only admissible for particular axes, and 

 the true form of the function from which it is derived will be that of a function 

 of the six variables used in my former memoir. 



It is, however, possible to obtain such a function of ( -j-, --^, -i ) as shall 



\aa; ay dzj 



not change its form with the transformation of co-ordinates. For, adding toge- 

 ther the values of these quantities, we shall banish (m', v' , w'), on account of the 

 relations, 



bc + b'c'+b"c"^0, 



ac -h a'c' -f a"c" = 0, 

 ab -h a'b' -f a"b" = 0. 



The result of the addition will be 



dx dy dz dx' dy' dz' 

 This quantity (w) will retain its value, independent of the directions of the 



