134 



Rev. S. Haughton on the Law of Fatigue. [June 15, 



This represents a cuspidal cubic ; and we are required to find values for 

 jS and A which will satisfy the experiments. 



2". The second method of applying the Law of Fatigue leads to an 

 equation which represents the experiments better than equation (2) ; and 

 the principle on which it is founded is probably a more correct applica- 

 tion of the Law of Fatigue. I assume that fatigue will occur when 

 the dynamical work multiplied by its rate, together with the statical 

 work multiplied by its rate, shall be constant ; or if WE represent the 

 total work of all kinds, and rate of both tvorks, then 



W E = E, + E, = constant (3) 



This is equivalent to assuming the total rate of work to be 



^^ W.B,+W.B . 



as in the problem of the specific gravity of a binary compound. 

 Equation (3) becomes at once 



= constant, 



or 



(iv-\-a).vn i- (iv -\- a) a; Bt n 



7 '- — = constant 



n t 



or finally, since (w-\-a) and a; are constants, 



(5) 



We have now to take equations (2) and (5) in succession, and find 

 which of them corresponds best to the observations. The method I have 

 followed is this : — Let any value 'of /3 be assumed, and substituted in 

 equations (2) and (5) for all the values of t ; the resulting values of A 

 will differ more or less from each other : let dA be the greatest differ- 

 ence between any two values, and let fj be the number of observations, 

 and S . A the sum of the values of A ; then I determine for each succes- 



and AA' is the double asymptote, corresponding to 



and having a cusp at negative infinity. There is a hyperbolic branch lying between 

 X' P and A' P. 



The portion of the curve with which we are concerned lies between OX and OY. 

 The number of lifts (n) attains a maximum MM' when 



l-l3t = 0, 



and the point of inflexion of the curve N occurs at double the preceding value of t • 

 after which the curve becomes asymptotic to the line OX. 



This equation represents a central czihic whose general form is shown in fig 2 

 (p. 135). It has a double point at infinity on the axis YY', which is a conjugate point 



