1882.] On Hypocycloids, Epicycloids, and Polyhedra. Ill 



l-(n-2)fz*+(n— 4)^ 3 -(w-6)^+ . . . 



= (\-nf Z z + ng?J-nKzt+ . . . )(t+i* 8 +i*H** 8 + ■ • • ) 

 where /, g, h are found to have the preceding values : 



2f=^ : 4*g = ^fn — 4-: and 67i would be ^gn—\fn+\. 



{^z is written in brief for 



4. For the continued product of the perpendiculars on a parallel 

 plane — 



<Pi+*)(Pi+*) • • • (p» + x) = (p + x)»- 1 ^LnaZ(p + xy~z+ . . . =U. 



The sums of the several powers of p l9 p 2 , . . . are found by taking 

 the logarithm, and differentiating on both sides. 



A X* l ^ X* 2 tf 3 ° U dx 



Xl XX 2 x s i 



3» s l l » 1.2 ^ ' * J 



By equating the coefficients of like powers of x, 

 vr; L X • 2 J 1.2.3.4 ' J 



where m is restricted not to exceed 5. 



Thus is established the first proposition (A) of § 1 . 



5. Proposition (B) of § 2 is proved as in § 4. 



The form, being universal, is equally applicable, when jp l5 p 2 , . . . 

 are used to denote the constants in the expression for distances, sucli 

 a,s — 



£ r 2 = a 2 _j_ v 2 _ 2a v COS X' 



Write in this form — 



