1882.] On Hypoeycloids, Epicycloids, and Polyhedra. 105 



Composition 

 of the 

 substance. 



Percentage 

 of Agl. 



0i • 









X. 









c. 







C. 











Agl 



100 -o 



142° 



156°*5 



-054389 + -0000372(T + 1) 



-0577 



6 "25 



Cu 2 I 2 .12AgI 



83 -1 



95 



228 



-05882 (from 16° to 89°) 



-0580 



8-31 



Cu.-Io.4AgI 



71 2 



180 



282 



-056526 + -0000410(T + t) 



-0702 



7 '95 



Cu.J.".3AgI 



65-0 



194 



280 



-059624 + -0000280 (T + t) 



0-0726 



7*74 



Cu 2 I 2 .2AgI 



55 3 



221 



298 



-061035 + • 0000295 (T + 1) 





7-88 



f„ T A rrT 



38-2 



ZOO 





n -nftQAQQ 4- n 'Annn^iVT -i- A 





8 •f^ l 7 

 o o / 



PbI 2 .AgI . . 



33 -8 



118 



144 



-047458 + • 0000026 (T + 1) 



-0567 



2*556 



In this table 6 1 and 6> 2 are the temperatures at which the structure 

 change commences and finishes, according to Mr. Rodwell's results, 

 ■c denotes the mean specific heat of the substance between t and T 

 lor temperatnres below ; q is the specific heat for temperatures 

 beyond 6> 2 ; and X is the heat absorbed by the unit weight of the sub- 

 stance in consequence of modification of structure. The relative 

 .accuracy of certain results and some probable conclusions are finally 

 discussed. 



VIII. " (I.) On a Tangential Property of Regular Hypoeycloids 

 and Epicycloids. (II.) On Theorems relating to the 

 Regular Polyhedra which are analogous to those of Dr. 

 Matthew Stewart on the Regular Polygons." By Henry 

 M. Jeffery, F.R.S. Received June 3, 1882. 



(Abstract.) 



I. On a Tangential Property of Regular Hypoeycloids and Epicycloids. 



1 . The following theorems will be established : — 



(A.) The product of the perpendiculars drawn from all the cusps 

 an each of these roulettes on any tangent is a function of the perpen- 

 dicular only, which is drawn on the same tangent from the centre of 

 the fixed circle, on which the roulette is generated. Consequently 



(B.) The sums of the cotangents of the angles which any tangent 

 to the roulette makes with the vectors drawn from the cusps to the 

 point of contact is a function of the cotangent of the angle made by 

 the vector drawn from the centre with the same tangent line, and of 

 the perpendicular drawn from the centre. 



2. These propositions are extended to spherical geometry, and their 

 dual forms stated both for planimetry and spherics. 



