OF DEFLECTIVE FORCES. i53 



the length of the curve, ^s, is expressed by the variation 

 of the ordinate y at any given point, reduced to the direc- 

 tion of the curve, and lessened by the length of a minute 

 curve of equal angular extent to the curve in question, 

 and of which the radius of curvature is equal to the varia- 

 tion ^y reduced to a direction perpendicular to the curve. 

 Now, in order to determine the shortest distance, we must 



put ^5=0, and — %=/d -^ gw. But at the beginning 

 d* d* 



and at the end of the line in question Sy must be =0, both 

 the points being fixed ; consequently the fluentyd -^ 8y 



=0, which can only happen when d t^=0, since 8y is not 



z=0, and the fluent cannot have difierent values, destroying 



each other, in different parts of the line, because the value 



must vanish equally for all parts of the line, which must be 



always the shortest distance between their extremities: 



d?/ 

 and the sine of the inclination -^ being constant, the curve 



as 



must become a right line. 



In the case of the present problem, we have Oi=Sf= 



r^j^ r^ ^^ .dxgda; + dySdy ^ dzgy 



simplify by making SdyziO and SyizO, confining the varia- 

 tion to dx, according to the spirit of the preceding de- 



monstration of the theorem; consequently mtzz- 



tts/ydz 



cix 

 = — -— -r- d^x ; and comparing this equation with the 



dy 



d^ 



dSy of the former example, we have in a similar manner 



