XJnd.ulalo)y Theory of Liglft, •4'27 



From Uiese equations we find 



SO that when 



{(T — d^) sin fl/ cos S/ + (t" (■cos'' 5/ — sin- 3/) = 0, 



the sum 2 . \|/ (r) A .r" A ?/ A ,jf is zero as required, while, of the 

 expressions for vf and v°- ^ one is a n>aximum and the other 

 a minimum. The last equation is always possible; for since 



• * / ,^ ' si" ^^/ . o /I - • o w ^ ., • - 



Sin 6/ cos l^ = ^ > ''i"<^* cos- 9/ — sm- fi, , = cos 2 9',, it gives 



cr' — cr 

 (16.) It has been observed (art. 1,) that the sums composed 

 of even products involving odd yiowers of the differences 

 must, in general, be very small compared with the sums com- 

 posed of products of the same degree in which the powers of 

 the diffeiences are all even. Let it then be supposed that 

 j;', ;/', ;:;' are rectangular coordinates of which the axes are 

 fixed in the medium; and that the arrangement of the niole- 

 ci'iles, with respect to these axes, is such that the sums of 

 which the terms involve odd powers of the differences A x^ , 

 Ay, Az', are either zero (art. 21,) or insensiblj' small. Let 

 the axis of Xj coincide with that of jr'; and let 6' be the angle 

 between y and i/i ; then 



Ax, = Ax' , 



At/i = Ay cos &' — A x' sin 6' , 



Azi = Ay sin 9'+ A z' cos 9'; 



and consequently, v.'hen we omit the terms involving the odd 

 powers of the differences A x' , Ay' , A z' , we have 



A .r/ A ,^;- = A x'- Ay'"- sin® 6' + A x"' A z'- cos^ 9'. 



For the reasons mentioned in article (9.) we leave out of 

 the expression for V/^ (art. 6.) the function $ (r), and then 



v/^ = cos= 9 . — S . v^ (;■) A z; Ax^ + sin^ 9 . -^S . vl/ (r) A r./ Ay^ 



— sin 9 cos 9 . ?« S . 4/ (r) A ,-/ Ay^ A x, . 



Now, when the coordinates jr^, t/^, z, are turned on the 

 common axis of x, and x' , the sum 2 . i|/ (;) A^;,^ A i/j^ must 

 be of the same value, whether ?/y coincide with y' or ;:'; we 

 will therefore suppose it to be sensibly the same for all values 

 of 9': when, again, we change the coordinates x^,y^,z,, 

 lor.r', y , .^', the last sum in the ecjuation will he composed 

 of terms involving odd powers of Ax', and will therefore, by 

 3E2 



