﻿1 — 





2b 



2b 





R 2 



(1 + 4%)* 

 26 





1 



1 



+ K 2 



_ r 2+% i 



li, 



— 6 Lci+4 



%)U 



422 Mr. A. Ferguson o;i Photographic 



il free " horizontal surface to the vertex of the drop. I^ is 

 the radius of curvature of the meridional curve at any point 

 P whose coordinates are (#, ?/), and R 2 , the second principal 

 radius of curvature, is measured by the length of the normal 

 at P to the meridional section intercepted between the point 



P and the axis of ?/. We have, therefore, where tan 6= -~- , 



y 2 sin (p y x 



With these data, it is easy to see under what circum- 

 stances the parabolic equation y = bj; 2 most nearly satisfies 

 the requirements of equation (a). For we have 



-p _(1 + 47At 2 )I (l + %)i 

 and 



Hence 



={46 + 8Zfy)(l — 6ty + 30ty . . . .) 



= 46-16^(1-1%+ . . . .) . . . . 09) 



= A-B^ 



if by and the succeeding terms — which, it should be noied, 

 are independent of the unit of length- — are small compared 

 with unity. 



A similar result can be obtained for the biquadratic 

 formula, y=&# 2 + c#*. This equation nrovides a nearer 

 approximation to the truth over a given range of the curve, 

 as, for a given value of y. the ratio of the second to the third 



term in the expansion of I . -f _ ) is sensibly greater than 



in the case of expansion (/3) above. 



Here also may be noted the effect of observational errors 

 on the quantities computed. Thus, putting 



and supposing z, b, and y to vary together, we have 



hz _ Bb 2{bhy + yhb) 6(b8y+y8b~) 



z ~ b + 1 + 1% ~ 1 + Uy ' 



