CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 155 



we see that 



d6 



. r dt (3 



cos -A = ■ = '— = const. 



v a 



where 4. is the inclination of the element rSd to the corresponding element 8s of 

 the brachistochrone. That is, the brachistochrone cuts all circles on the above 

 sphere, whose planes are parallel to xy, at a constant angle. {Loxodrome.) 



11. It is easily seen that 



T = C 



is the equation of an Isochronous surface. 

 Also, since 



fdr\ fdr\ fdr\ 



\dx J \dy / \dz ) 



dx dy dz 



dt dt dt 



the brachistochrone cuts all such surfaces at right angles. 



And the normal distance between two consecutive isochronous surfaces is 

 proportional to the velocity in the brachistochrone of which it forms an element. 

 For, of course, 



8s=VOT. 



12. Generally, putting 



_ /dr\ 2 /dr\ 2 /dr\ 2 1 



* = (dx) + (dy) + U) = 2(H=T) ' (7). 



1 



(8), 



we have 



and x _ 



with similar expressions for Y and Z. 

 Also, by (1), we have 



2(H- 



V): 



_ ® ' 





\dso . 



) = 



1 



2W 



d® 

 dx 











dT _ 

 dx 



UX ,, \ 



and 









dT 



dE ~ ' 



- r®dt j 



Hence 







d 2 x 

 dt 2 



_d n dT\ 



~ dt\®dx) 



__ 1 d /dr\ 

 ~Wdt \dx) 



1 dTdM> 

 ® 2 dx dt * 



VOL. 



xxrv. 



PART I 









(9), 



(10). 



2u 



