388 Proceedings of the Royal Irish Academy. 



stationary for synall arbitrary and discontinuous variations of the 

 higher fluxions, gives us the conditions in (3) at once, as stated in 

 the note to eg^uation (18). Hence, in Ai^, the only terms remaining 

 are those corresponding to Yf^ab) ^^{ab), *-^- to Y^b {i/"^^ - 2/*^"*^). 



Hence it is evident that the condition -E" > is necessary ; for <t 

 may be taken for any small portion of the integration. 



To show that it is sufficient, it is only necessary to observe that 

 we may extend o- to include any large portion of, or the entire of, the 

 integral. 



§ 10. It will be observed that when some of the variations are large, 

 the fact that E is always positive ensures that the integral shall be a 

 minimum, even though the condition relating to the " conjugate point" 

 in small variations be not fulfilled. 



Thus an arc of a great circle on a sphere is a minimum compared 

 to all neighbouring lines for which the direction is, for finite lengths, 

 inclined Q.t finite angles to the direction of the great circle, the distance 

 between the two lines being always indefinitely small, and this property 

 evidently holds when the arc of the great circle is greater than a semi- 

 circle. 



§ 11. It is interesting to observe that we cannot derive the condi- 

 tion for a maximum when the variations are small from the form 

 assumed by the condition -E" = when the varations are small. 



§ 12. It may be well to give some examples. 



1. The brachistochrone. — Here 



?7 = 



so that 



Jl +y- , 



;= — dx. 



1 



y 



Jl + ?/"- Jl +y^ 



Jy Jl+/ 



an expression which is obviously positive, when y and y are different, 

 and when the square roots are taken with positive signs. 



In this case, if we give a variation which makes dx negative, we 

 must, in order to get the time of the descent, change the sign of one 

 of the square roots, and we still get £dx positive. 



