164 



Mr Bateman, The determination of curves 



When we add the small element QQ' the new centre of gravity 

 will in the limit lie in GQ: therefore GQ is the tangent at G to 

 the locus of G. 



Thus the equations (1) are those of a curve which passes 

 through an arbitrary point 0, and whose tangent always meets 

 the given curve. 



They can also be written in the form 



« doc . \ 



as 



%w + u(%-x)=j u~ 

 7}W + u (?7 — y) = j 



•(2) 



Q dy , 

 u -f- as 

 p as 



u 

 accordingly, if we produce GQ to R so that GR — — GQ, the co- 



1 f dx 

 ordinates of R will be — Yu^-ds, etc.: and therefore R describes 

 w J as 



a curve having the same tangent indicatrix as the original curve. 



Consider now a group of curves having the same tangent 



indicatrix. Starting with one curve, the equations of any other 



curve of the group are of the form 



CtJb -m 



x = \u-rj-as 

 as 



y' 



u -^ ds 

 ds 



dz , 

 z' = \u -j- as 

 ds 



.(3). 



It is easy to verify that for the new curve (with the usual 

 notation) 



p = pu, a' ' = cru, ds' = u ds, dx = u dx. 



