Involutes to a Circle. 297 



Mr. Todbunter in the later editions of bis ( Integral Calculus,' 

 accompanied with a reference to another English treatise, from 



the tangents at P, P', draw perpendiculars from an arbitrary point O, and 

 we obtain at once, by inspection, 



dp = - q8(p, 8q + Ss==pS#, 

 whence 



1 dtf F ^ dcf> 2 d(f>' 

 Or, again, proceeding analytically, we have 



x — A= fdscoscf), y — B=fds.sm<f); 

 whence, integrating by parts, 



x — A = G cos <f) — G ' sin 0, 

 y — B =Gsin$+G'cos</>, 

 where 



G=s'-s'"...; G'=s" -«""... j 

 whence 



r=G 2 + G' 2 and G+G" = -§. 



d(p 



From which also we may deduce 



CUT r~{t o 9 o /-io 



q = r— -=G, w-=r- — <^ = (jr. 



ds 



This last demonstration would at first sight seem to be only valid for the case 



ds 

 of the G series coming to an end, i. e. of -y-p being a rational integral function 



d(p 

 in <f> ', but it would be quite legitimate to infer at once from it the univer- 

 sality of the equations above written connecting r 2 with 5 and (p ; for we 

 may write down the general differential equation of the second order 



dcos-i~-d6=d<j>, 

 ds 

 i. e. 



dr 



d. 



V(dsy 2 - (dr) 



ds V ds 2 — dr 2 j , 

 — dep, 



in which —^ mav be considered as given, and r or r 2 to be determined. 

 dep 



The equations in question, having been shown to be true for a form — con- 



d<p 

 taining an indefinite number of arbitrary constants, evidently can only 

 amount to a transformation of, and may be used in supersession of the 

 equation last written. It may be worth while to set out this latter under 

 a more familiar form of notation. If, then, we use y for r and x for </>, 



ds 

 and call — =X (any function of a?), it becomes 

 dep 



-xy+xy Vx^y 2 



XVX 2 -*/' 2 y ' 



an apparently very complicated form of equation, but admitting of the 

 simple solution y 2 =u 2 +u' 2 , where u satisfies the linear differential equa- 

 tion u-\-u"=K. 



