1915-16.] The “ Greometria Organica” of Colin Maclanrin. 145 
§ 60. Differential Geometry of Pedals and Maclaurins Theorem for the 
Curves pjr = (r/a) n . 
Use of the p — r Equation. 
Let, as usual, 
p, r ; Pv r i ; Pv r 2 > etc -> 
denote corresponding elements of the curve and its pedals. 
Then 
p/r = V\l r v = pf r 2 = etc. . 
and 
r m = p m /r m ~ 1 j p m = p m+1 / r m . 
( 1 ) 
( 2 ) 
Let S, Sp . . . S w be corresponding arcs of the curve and its pedals, and 
let <t> be the angle between a radius vector and the corresponding tangent. 
Then, up to sign 
ds = dr sec = rdr/ff r 2 - p 2 
ds 1 = drj sec </> = rdpjff r 2 - p 2 
ds 0 = dr 0 sec </> = -^ds ' 
ds m — 2—ds m _ 1 — — ds m _2 
(3) 
Cor. 1. 
dsjds = dp j dr. 
Cor. 2. The elimination of p/r from (3) gives rise to a variety of results, 
e.g. from the equivalents of ds 2 'and ds 3 in (3) we deduce 
ds Y 
ds 2 
v 
ds 
ds ± 
ds ds 2 
ds 2 
ds s 
A 
ds 1 
ds 2 
ds 1 ds z 
Also 
Let the tangent 
Then 
ds k ds k+1 ds k _ |_ 2 
dsi ds l+1 ds l+2 
ds m - (-2 
ds m ds m . |-i 
= 0 
PP 1 = * 
• • LiL 2 = ^i 
P 2 P 3 = ^ 2 , etc. 
(4) 
( 5 ) 
VOL. XXXVI. 
t = ffr 2 - p 2 
t^WP-p 
, n 2 
P A 
etc. 
( 6 ) 
10 
