108 Mr. H. M. Jeffery. [June 15, 



+( i )V , { ^i_ ( ,_ 2)fer} 



_^»H»/ ^--4)^ -5)_ 0^2)(n-5) ^ w J-« | + _ 



The series ends with, the term involving jp s or p 4 , according as w. is 

 odd or even, as may be seen by examining the expansion of cos nO in 

 an ascending series. 



8. The eqnation to the regular epicycloid is 



sinx= V To\ » where ( n + 2 )x= n (^+ e \ 

 Hence the formula for this epicycloid is 



i» • fer»--©*'-{<"«>fe)"-'} 



I + 2)0-2)^-3) / n y-^ rcQ-g) 1 

 W I 1.2.3 \n+W 1.2 J" 1 " ' * ' 



By reversing the order in expanding cosw#, it appears that no 

 terms involving^ 3 , p, p° occur. 

 12. Proof of Stewart's theorems. 



Since the formulae of § 6 are general, they apply to a parallel line, 

 on which the perpendiculars are drawn. 



<Pl+*)(P* + x ) • • • (lh + X ) 



The sums of the several powers of p 2 , . . . are obtained by taking 

 the logarithm, and differentiating on both sides — 



x x* 1 a> 3 2 a?* 3 U dx 



(This last quotient is remarkable,' as it shows the matrix out of which 

 these theorems of Dr. Stewart arose.) 



= ^{i+?(»Y_ 1 _ + i^(«Y_ 1 _+ ... "I 



p + A l\2/(p + ii) s 1.2W(j) + i)' / 



