8 



are obtained by putting v under the form of a homogeneous function 

 of a, ^, 7, of the first dimension, with the help of the relation 

 a* + ^* + 7* — 1, and by then differentiating this homogeneous 

 function, as if a, (B, y, were three independent variables ; finally, the 

 definite integral fvds is taken from the luminous origin to the point 

 X, y, z, and the variation dfvds is obtained by supposing the co- 

 ordinates of this last point to receive any infinitely small changes, 

 the colour remaining the same. 



2. To deduce the equation (/i) from the known condition of least 

 action, let us observe that by the calculus of variations, 



ij'vds =y(S». ds -j- V. ids) ; 



in which, by what we have laid down respecting the form of v, 



^^ = 3^^* + 1]^^^ + 5^^" + 3T ^" + 37 ^^ + S?^^' 

 3w 3(' , 3o 



*' = ''3«+^3j + ^37' 



while, by the nature of a, /S, y, 



ia~ ds 4* »• ids = 3. ads ^ i. dx ^ d. ijTt 

 3/8. ds + /3. "ids = 3. fids = i. dy z= d. iy, 

 3y. ds -j- y. ids = 3. yrf* = 3. «/z = rf. 3z ; 



we have therefore, 



'/"" =/( ff " + 1 '^ + 5^ '0 ■" +/( L- ■«' + 1 ^" •<- g ■"-) 



Ju . 3c'. , . 3d » 3u' - , , Jv . 3u' 



= -— 3x — ^— ,3x'+ »-3y — Tr,3u'+ v- 3z — s— , 3z' 



3« 3» "^JiS * J;3' ' ' 3y 3y' 



3y 



the accented quantities belonging to the first limit of integral, and 

 disappearing when that limit is fixed. The condition of least action 

 requires that the quantities which remain under the integral sign, as 



