38 Prof. Challis on a new Integration of Differential 



Again, 



because jop' + 1 = 0. Hence 



dx= _ % {y ^ )+ d J i= _ % . fcO + % 



piz \j J J * pi pi2 2a — p p 



But 



and 



v 



£ =dp= „(l±i& dx ., 



Consequently, by substituting, 



y n [p-aY p 



Now 



**- f =( l + f) dx = -i^ dx > 



because dp 2 =zd%' 2 -\-dy 12 . Hence it will be readily seen that 



<¥ _ (p—a)dp 

 y' p{2a-p) 



This equation gives, by integration, p 2 — 2ap= — ky n } A- being an 

 arbitrary constant; and by solving the quadratic, 



p = a + v'a? — ky H . 

 Then, since 



it follows that 



*.*#{£££*}<. 



(S) 



which is the differential equation of the evolute. If b and c be 

 respectively the values of?/ which satisfy the equations a 2 —ky t2 =0 

 and (k 2 + k)y 12 — a 2 = 0, the equation may be put under the form 



Hence it is evident that the values of y' all lie either between 

 + b and -fc, or between —b and —c. The least value c of?/ 



belongs to a cusp, because it makes -—j infinite ; also the curve 



