the Equation of Fresnel's Wave, 383 



5. Expanding it first under the form 



w 8 + W 10 + W 12 = 0, 

 in which W 8 , W 10 , W 12 are, respectively, homogeneous func- 

 tions of x, y, z, of the 8th, 10th, and 12th dimensions, we 

 soon discover that these three functions have a common factor, 

 of the 8th dimension, namely, c 2 z 2 r 2 R 2 , in which 

 R 2 = C 2 + 4 a 2 U 2 (c 2 -a 2 ) (c 2 -^) x 2 y\ 

 C having the same meaning as in (I.), so that 



R 2 > 0, if c 2 > b 2 > a 2 , or if c 2 < V 2 < a\ 

 conditions which we may suppose to be satisfied . And reject- 

 ing, as evidently foreign to the question, this common factor 

 c 2 * 2 r 2 R 2 , the known equation of the wave results, under the 

 form 



«o + U i + u 4 =» 0> (IV.) 



in which u = a 2 b 2 c% 



u x = - {a 2 (b* + c 2 ) x 2 + b 2 (c 2 + a 2 ) y 2 + c 2 {a 2 + b 2 ) z 2 }, 

 w 2 = (x' 2 +y 2 + z' 2 ) {a 2 x 2 + b 2 y 2 + c 2 z 2 ). 



6. The foregoing investigation is taken from a manuscript 

 Report which I had the honour of drawing up in July 1830, 

 when, in conjunction with the late and present Provosts of 

 Trinity College, Dublin, I was appointed to examine the first 

 communication of Professor MacCullagh to the Royal Irish 

 Academy, since published in the second part of the sixteenth 

 volume of the Transactions of that body. A far more con- 

 cise and elegant deduction of the same, known equation of the 

 wave from the same geometrical construction, depending, 

 however, a little more on the geometrical properties of the 

 ellipsoid, has since been communicated by Professor Mac- 

 Cullagh himself, and is published in the second part of the 

 seventeenth volume of the Transactions of the same Academy. 

 Others have published other demonstrations. 



My own mode of deducing the equation of the wave from 

 the principles of Fresnel, without any reference to the ellip- 

 soid above referred to, may be seen in the ' Third Supple- 

 ment' to my Theory of Systems of Rays contained in the first 

 part of the last-mentioned volume. 



Observatory of Trinity College, Dublin, 

 October 13, 1841. 



P.S. Since writing out and sending off the foregoing paper, I have had 

 opportunity to refer to FresnePs own deduction of the*same equation of 

 his wave from the same geometrical construction, entitled " Calcul tres 

 simple qui conduit de f equation d'un ellipsoicle a celle de la surface des 

 ondes." (Mem. de VAcad. des Sci. de Flnst. Royale de France, torn, vii., 

 page 137.) It is much simpler than mine, and nearly coincident with 

 that of Professor MacCullagh, but seems to have been overlooked by both 

 of us. 



