.3W Mr. Challis on the Theory of the 



limit. Therefore also the values of , J must range below a 

 certain small limit. And since 



dtdx ' dx' 



if the condensations be given by 



. ttS 



V = m\ sin — , 

 the accelerative force corresponding to the condensation ^, varies as 



dy ttx 



-^ , or MiTT cos — , 

 dx \ 



and is greatest when 



x = 0, and -— = ottt. 

 dx 



Hence m can never exceed a certain small quantity. Moreover 



as the condensation as = — —~- , and that the greatest value of 



adt '^ 



y is mX, it follows that m\ must not exceed a certain limit. 

 But the value of \ is not limited. These considerations exclude 

 those forms of the primary curve in which the maximum or- 

 dinate is not very small compared with x. In general when 

 the motion is given by an irregular line, the angle which two 

 consecutive tangents to it make with each other, must not exceed 

 the greatest value m-K admits of, and the ordinates to the line 

 must not surpass the limiting value of m\. If x be supposed 

 very great, and 



X ^ . /TT irSSX ^ irZ 



x= - ±z, y = m\ sin I - ± — I = m\ cos -— = mX 



2 v2 X / A. 



very nearly, shewing that a portion of the curve becomes a 

 Jitraight line parallel to the axis. Hence the motion indicated 

 by this line is possible. Also when X is very great, y = irmx, and 



