152 OF DEFLECTIVE FORCES. 



Scholium 1. The demonstration implies that the ori- 

 gin of the curve must coincide with the uppermost given 

 point : now only one cycloid can fulfil this condition and 

 pass through the other point, and it will often happen that 

 the curve must descend below the second point, and rise 

 again. 



Scholium 2. The method of independent variations 

 may be applied with great elegance and simplicity to pro- 

 blems of this kind, although it has too commonly been made 

 complicated and perplexed by unnecessary abstraction. 

 An example of its application has already occurred in the 

 investigation of the properties o( Jvds (266), but it will not 

 be superfluous to enter into some further illustration of 

 the method on this occasion. 



Let it be required, for example, to determine the equa- 

 tion of the line which gives the shortest distance between 

 two points, from the property of maximums and mini- 

 mums which are unaltered by any slight variation of their 

 elements. We have, therefore, SsziO; but ds=:Jd^s, the 

 characteristic y^relating to the fluxional variation expressed 

 by d ; and J^d^szuJ^Ms (265), Now, x and y being the 

 ordinates, and s the curve, we have ds^zzd^r^ + dy^, and 



Sd5- T — ^—^; and, for the sake of simplicity, we 



may make ^domO, supposing the curve to pass into a 

 neighbouring form by the variation of dy only : we have, 



then, 8d5=:-T^ 8dy, of which we must find the fluent. Now 



^(S^3.)=g%^dg..=|% + dg.,; eonse- 

 qnently f Ms zz^^y—fd -^ ^yziSs. This expression im- 

 plies, when geometrically considered, that the variation of 



