Mr. I. Todhunter on a Problem in the Calculus of Variations. 425 



19. The following stars have also been observed : numerous 

 lines are seen in the spectrum of each • and in some, several of 

 the lines were measured ; but we have not yet instituted any com- 

 parisons with the metallic spectra. 



Castor ; e, £ and rj Ursa majoris ; ct and e Pegasi ; a, /3, and 

 7 Andromedce, the last an interesting spectrum ; Rigel, a spectrum 

 full of fine lines; tj Ononis ; a Trianguli; y and e Cygni; a, J3, 

 y, e, and rj Cassiopeia ; <y Geminorum ; Canis majoris ; /3 Canis 

 minoris; Spica ; y, 8, and e Virginis ; uAquilcs; Cor Caroli ; 

 /3 Auriga ; Regulus ; /3, <y } h, e, £ and rj Leonis. 

 [To be continued.] 



LXII. On a Problem in the Calculus of Variations. 

 By I. Todhunter, M.A., F.R.S.* 



THE problem to which the following remarks relate may be 

 thus enunciated : — To determine the greatest solid of revo- 

 lution, the surface of which is given, and which cuts the axis at 

 two fixed points. The problem has been discussed in the Philo- 

 sophical Magazine for July 1861, August 1861, September 

 1862, and March 1866; but something may, I think, be added 

 to what has been already published. 



Adopting the usual notation, we have to make 7r\y 2 dx a 



maximum, while 2it §y \/(l -i-p q )dx is given. Let a be a con- 

 stant at present undetermined ; then we have by the received 

 theory to make u a maximum, where u denotes 



${y* + 2ayS(l+p*)\dx. 

 We obtain 



8u = f A[$y —p$x)dx -t- B, 

 where A stands for 



and B stands for certain terms which are free from the integral 

 sign. 



The expressions for u and Su are both supposed to be taken 

 between limits which correspond to the fixed points on the axis. 



Now by the known principles of the subject we put A = 0; 

 this leads in the usual way to 



2a y -A-jf, 





•.(■l.+JP*) 



* Communicated by the Author. 



