TRANSACTIONS OF SECTION A. 



453 



Three linear relations, it is known, determine an infinite number of such pairs, 

 whicli lie in general on one cubic curve. The equation to this cubic is 



.V l^ l.^ /j 



I 



■ If, however, two of the relations be such that the conic passes through two 

 fixed points (.r,, y^, ~i){.V2, y^, So) the cubic contains as a factor the line joining 

 these points. 



Hence we obtain the identity 



r .fj' 

 y y; 



y^: 



I 

 m 



z Sj* Sj^ n 

 z y 2yja, 2^58^ 2p 

 z .V 2zyi\ 2z^x„ 2q 

 y .V o 2.i\y^ 2.v^y..^ 2r 

 where Xj, X„, &c., are the minors of x\, a.:,, &c., in the first factor. 



■ 2{'^•,yl,^a} {Ix^^^ + &c +ij(y,S2 + s^yj + &c.} 



7. On the Bectijia.hle Splierical Epicycloid, or Involute of a Small Circle. 

 By Henry M. Jeffery, F.B.S. 



1. As in plane geometry the involute of a circle is a terminal form of an 

 epicycloid, so in spherics if a great circle roll on a small fixed circle, a point in its 

 circumference will be the involute of the latter. This involute is rectifiable, as 

 has been shown by Clairant and J. Beruouilli, and is its own polar curve. 



2. In cycloidal and trochoidal curves the radius of curvature and the evolutes 

 are conveniently expressed in terms of the perpendicular and radius vector. 



Let p, r be the co-ordinates of the curve p', r' those of its evolute, p, p' their 

 radii of curvature. 



If p -f (»•) denote the curve, then as in planimetry, sin^ p' = sin- r — sin* 2>, and 



. dd 



sm r --- , cos ?• 

 sm r as 



sm p 

 since — . ' 



= cos r cos p + sin jj sm p. 



In differentiating, p, r' remain constant, while r, p vary. 



. sin ?• dr 



tan p = 



cos p dp' 



(Mathematical Tripos Examination, .Tan 17, 1882.) 



By eliminating from these four equations, the equation to the evolute, or con- 

 versely to the involute, may be obtained. 



3. In the small circle p' = r' = a: heuce the involute of the circle is defined by 

 the equation 



sin-/- — &vol'p = sin-rt. 



Since the equation is unaltered, when 



and 



2 



■r, are substituted for r 



and p, this epicyloid is its own polar curve. 



4. To find the involute of the involute of a small circle. 

 By § 2 the relations of the co-ordinates are 



cos ;•' = cos r cos J? + sin p sin R, where tan R = ' — , sin- «' = sin- ;• - sin- p. 



cos p dp 



In this epicycloid by § 3, cos* r' + sin- p' = cos- a. 



The diflerential equation to its involute is 



sin* a = cos* r + sin* p — (cos r cos R + sin p sin R)". 



To separate the variables, let p cos 6 = cos r, p sin ^ = sin p. 



p' {d 6)" = (p* - sin* a){{d p)* + p- (d 6)"-} 



