2 Prof. Powell on the Demonstration of Fresnel's Formulas 



ing on assumptions in some instances confessedly empirical, in 

 others dependent on analogies, or hypotheses not free from doubt, 

 and at any rate little connected into a system. 



4. This investigation, whose questionable points are so fairly 

 stated, and ably grappled with, by Mr. Airy in his tract on the 

 Undulatory Theory (1831, art. 128 et seq.), has been since pur- 

 sued on different principles by M. Cauchy, and especially by the 

 late Prof. Maccullagh in his memoir ^' On the Laws of Crystal- 

 line Reflexion and Refraction^' (Mem. Roy. Irish Acad. vol. xviii. 

 1838), whose views have been ably but briefly expounded by 

 Dr. Lloyd in his Lectures on the Wave Theory (part 2. p. 30, 

 1841). More recently, Mr. Power has investigated the subject 

 by a systematic analysis, directed to other objects, but including 

 an important element in these deductions (" On Absorption of 

 Rays,'' &c., Phil. Trans. 1854, part 1). 



5. But among these distinguished philosophers there exists 

 considerable diversity, and even contradiction of views. Nor, so 

 far as I am aware, has the subject been so discussed as to enable 

 us to trace the source of these discrepancies, or fairly to estimate 

 the claims of the opposing theories, or the force of the experi- 

 mental results which bear upon them. Thus it seems highly 

 desirable, that questions affecting so fundamental a part of the 

 undulatory theory should be cleared up and placed on an un- 

 assailable basis. 



Having long ago thrown aside some investigations on the 

 subject, in which I was then engaged, I have of late had my 

 attention recalled to the question, and have thus been induced 

 to revise and extend those investigations, in the hope of contri- 

 buting towards the settlement of the points involved, or at any 

 rate of putting the whole discussion before the student in a con- 

 nected point of view ; with which object I have been led to com- 

 mence ab initio, so that those who have only an elementary 

 acquaintance with the theory may be enabled to follow the de- 

 ductions without difficulty, and may here be furnished with that 

 systematic elucidation which is not, as far as I am aware, to be 

 found in any existing publication. 



Theoretical Views, 



6. The formulas for the amplitudes of the incident, reflected, 

 and refracted rays, as given by Prof. Maccullagh and later writers, 

 though closely corresponding with those of Fresnel, and fulfilling 

 generally the same conditions, yet differ from them in certain 

 cases as to the sign, and in others as to the values of the ex- 

 pressions. 



7. But the main point of difference and difficulty consists in 

 this : Fresnel investigates two sets of formulas ; one set (H) for 



