522 MR CLERK MAXWELL ON THE THEORY OF ROLLIKG CURVES. 



the letters «i s., «,, to denote the length of the curve from the pole, jh p^ p-i for the 

 jierpendicnlars from the pole on the tangent, and qi q-, q^ for the intercepted part 

 of the tangent. 



Between these quantities, we have the following equations : — 



J- = r cos S 



O 



IT 



di 



r d r 



9 = 



{■^^ my 



R= 



d r\- (P r 



-C-I) 



We come now to consider the three equations of rolling which are involved 

 in the enunciation. Since the second curve rolls upon the first without slifpiwj. 

 the leno-th of the fixed curve at the point of contact is the measure of the length 

 of the rolled curve, therefore we have the following equation to connect the fixed 

 curve and the rolled curve, — 



Now, by combining this equation with the two equations 



it is evident that from any of the four quantities Q^ i\ 6, r, or*, 3/, .*■, ;/_„ we can 

 obtain the other three, therefore we may consider these quantities as known func- 

 tions of each other. 



Since the curve rolls on the fixed curve, they must have a common tangent. 



Let PA be this tangent, draw BP, CQ perpendicular to PA, produce CQ, and 

 draw BR perpendicular to it, then we have CA=*„ BA=( „ and GQ=r^ ; CQ=7',, 

 PB=;a,, and ^l^=p,; AQ.=q„ AP='/, and CN-^,- 



I 



