11. Incestigation of the Equation to FresneVs Wave Surface. Bii 

 Archibald Smith, Esq., Trinitij College, Cambridge. 



[Read May 18, 1835.] 



" The mathematical difficulties under which the beautiful and in- 

 teresting theory of Fresnel has hitlierto laboured arc well known, and 

 have been regarded as almost insuperable. He tells us in his Memoir 

 (see the Memoirs of the Royal Academy of Sciences of Paris, torn. vii. 

 p. 136.) that the calculations by which he assured himself of the truth 

 of his construction for finding the surface of tlie wave were so te- 

 dious and embaiTassing, that he was obliged to omit them altogether. 

 A direct demonstration has since been supplied by M. Ampere {An- 

 nales de Chimie et de Physique, tom. xxxix. p. 113.); but his solution 

 is excessively complicated and difficult." A geometrical demonstration 

 of considerable simplicity has been given by Mr M" Cullagh in a paj)er 

 in the xvi'" Volume of the Transactions of the Royal Irish Academv. 

 from which the preceding paragraph has been quoted. 



The difficulties whicli were experienced in this problem arose from 

 two causes of the same nature : — want of symmetry in the funda- 

 mental equations, and the use of the essentially unsymmetrical method 

 of differential coefficients. By putting the fundamental equations of 

 Fresnel under a symmetrical form, and by the use of the Metliod 

 of Multipliers as it is employed in the M^canique Analytique, the 

 eliminations may be effected without difficulty. 



To render what follows more intelligible, and to show in what it 

 <iiffers from tlie other methods, I shall give the fundamental equations 



