TRANSACTIONS OF SECTION A, 499 



same rule as the curvature of rays ; that is to say, its value at each point must be 

 equal to the rate at which the logarithm of the velocity changes as we travel along 

 the principal normal. For a free path the velocity increases inwards, and for a 

 brachistochrone outwards. 



Tiie following proof of this law of ray-curvature rests on the physical principle 

 that rays from a point are cut orthogonally by surfaces of equal time. 



_ Let T denote the time light taifes to travel from a given fi.\ed point to any 

 point. Then, if ds be measured along a ray, we have for the difference of time to 

 its two ends 



dT = !^, 



V 



V denoting the velocity of light ; and if d.v be measured at an inclination a to a ray, 

 the diflerence of time to its two ends is 



rj, _ dx cos g 



V 



Hence 



dT_co3a 

 dx V ' 

 and similarly 



dl' _ sin a 

 dy ~v~ 

 Therefore 



d cos a _ d sin a 

 • • • • dy V d.v V 



Performing the differentiations indicated in this last equation, and then putting 

 <i = 0, so that the axis of x is tangential and the axis of y normal, we have 



da 1 do d t d , 



J- = r- = --=- log 2; = — . log ti : 



d.v V dy dy ° dy ° '^ 



■where — is the curvature — , and — expresses rate of variation along the 



ax p ay ° 



normal. 



Several of the foregoing results have been previously obtained by the employ- 

 ment of Maupertuis' principle of Least Action ; but to many students the optical 

 proofs here given will be more intelligible. 



Reference may be made to a special method of deducing brachistochrones from 

 free paths by Professor Townsend (' Quart. Journ. Math.' vol. xiv.), and to a paper 

 by Professor Larmor on Least Action (' Proc. Lond. Math. Soc' vol. xv.). 



4. On Curves in Space. By Professor Caylet, F.B.S. 



5. On the Extension and Bending of Cylindrical Shells. 

 By A. B. Basset, M.A., F.B.S. 



The recent investigations of Lord Rayleigh' and Mr. Love" have directed 

 attention to this subject, and I propose in the present paper to discuss two points 

 connected with this question. 



The potential energy of a shell of thickness h consists of two terras, one of 

 which is proportional to /i, and depends upon the stretching of the middle surface, 

 whilst the other, which is proportional to k', depends upon the bending. The theorv 

 which has been adopted by most English writers upon the vibrations of thin 

 shells, supposes that the energy due to bending is the most important, and that the 



' Proc. lioy. Soc, vol. xlv., pp. 105 and 413. 



■-■ Phil. Trails., 1888, p. 491; and Proc. Lond. Math. Soc, vol. xx., p. 89. 



K K 2 



