332 A. Mukhopadliyay — Differential Fjqnation of all Parabolas. [No. 4, 



and as, moreover, in every parabola, the reciprocal of R vanishes, tho 

 differential equation of all parabolas in terms of p and w is 



To integrate this, put 



/udoi 



whence 



or, 



which gives 

 so that 



and 



p = e 



1;^ = "' + '' 



_ Sdu 



dm = 



^2 + 9 ' 

 -w = 3 tan (o) + !<) , 



I ?^cZw = 3 1 tan (o) + 7.:) t^w 

 = 3 log m sec (o) + 7i;) , 



/■ 



p = e = m^ sec^ (w + 7^) , 



which, therefore, is tlie relation between p and w in every parabola, lead- 

 ing at once to the intrinsic equation 



s = m^ sec^ (w -^ Jc) doi f 



and, if the origin be suitably chosen, we may put A; = 0, so that we 



have the well-known result 



o r dm 

 s = m^ / 5- . 



J COS'^Ot) 



14^7^ May, 1888. 



* See also P. A. S. B. 1888, p. 84, footnote. 



