278 Mr. Challis on the Theory of the 



a point may be taken in the axis, such that the ordinates at equal 

 distances on each side of it, are equal with opposite signs. At 

 this jjoint, the value of the ordinate is o, because if z be made 

 equal to zero in the equation, F{1' -z) = - F{1' + z), F{l') = -F{l], 

 which cannot be, unless F{l') = o. Also at the point where F{l—z) 

 = F{1 + z), the ordinate is a maximum or minimum ; for — F' {l — z) 

 = F'{l + z), and making z = o, F' {l)= —F' (l), which cannot be, 

 unless F' [I) =. o. Hence I' and I are so related, that when they 

 are measured from the same origin, the difference between them, 

 is always equal to the interval between a maximum ordinate, 

 and a point where the ordinate = o. The curve which satisfies 

 all the above conditions, is described in Fig. 1. It must extend 

 indefinitely in both directions, because it must be equally dis- 

 posed on each side of the maximum ordinate OQ, and on each 

 side of C, the point where the curve cuts the axis. ^B, BC, 

 CD, &c. are all equal, and the portions AaB, BbC, CcD, &c. 

 are all exactly alike. Let AB = \. This is the only parameter 

 which the equation of the curve can contain. Hence 



y = F{z)=X.F{l). 



Also because the curve extends indefinitely in the positive and 



negative directions, ^Tt y must be some trigonometrical function 



of the abscissa, considered as an arc of a circle, the circumference 

 of which is 2X. The simplest that presents itself is 



y = m\ sin ~, 



m being an arbitrary numerical quantity. The required condi- 

 tions will also be satisfied in the most general manner by the 

 equation 



. ttZ , . SttZ „ . birZ . 



y = OT\ Sin --4- m>. sin -r 1- m \ sm --— + «c. 



X A A 



