On the Imnar-diumal Magnetic Variation at Toronto. 459 



producing in every limar day a variation which is distinctly appre- 

 ciable, in each of the three elements, by the instruments adopted and 

 recommended in the Report of the Committee of Physics of the 

 Royal Society, when due care is taken m conducting the observa- 

 tions, and suitable methods are employed in elaborating the results. 



2. That the lunar diurnal variation in each of the three elements 

 constitutes a double progression in each lunar day ; the declination 

 having two easterly and two westerly maxima, and the inclination 

 and total force each two maxima and two minima between two suc- 

 cessive passages of the moon over the astronomical meridian ; the 

 variation passing in every case four times through zero in the lunar 

 day. The approximate range of the lunar-diurnal variation at To- 

 ronto is 38" in the declination, 4"-5 in the inclination, and "000012 

 part of a total force. 



3. That the lunar-diurnal variation thus obtained appears to be 

 consistent with the hypothesis that the moon's magnetism is, in 

 great part at least if not wholly, derived by induction from the 

 magnetism of the earth. 



4. That there is no appearance in the lunar-diurnal variation of 

 the decennial period, which constitutes so marked a feature in the 

 solar diurnal variations. 



"On Autopolar Polyedra." By the Rev. Thomas P. Kirkman, 

 M.A. 



An autopolar polyedron is such, that any type or description that 

 can be given of it remains unaltered, when summits are put for faces, 

 and faces for summits. To every /3-gon B in it corresponds a /3-ace 

 b (or summit b of /3 edges), which may be called the pole of that 

 ^-gon ; and to every edge AB, between the a-gon A and the /3-gon 

 B, corresponds an edge ab, between the a-ace a and the /3-ace b. 

 Two such edges are called a gamic pair, or pair of g amies. 



The enumeration of autopolar ^-edra is here entered upon as a 

 step towards the determination of the number of ^j-edra. The 

 theorems following are established, and shown to be of importance 

 for the solution of the general problem. 



Theorem I. — No polyedron, not a pyramid, has every edge both 

 in a triangle and in a triace. 



Bef. An edge of a polyedron is said to convanesce, when its two 

 summits run into one ; and it is said to evanesce, when its two faces 

 revolve into one. 



An edge (AB) is said to be convanescible, when neither of the 

 faces A and B is a triangle, and (AB) joins two summits which have 

 not two collateral faces, one in either summit, besides A and B. 



An edge {ab) is said to be evanescible, when neither a nor 6 is a 

 triace, and the two faces about {ab) are not, one in either, in two 

 collateral summits, besides a and b. 



Theorem II. — Every polyedron, not a pyramid, has either a con- 

 vanescible or an evanescible edge. 



Theorem III. — Any p-edral q-acron, not a pyramid, can be 

 reduced by the vanishing of an edge, either to a (p — \)-edral q-acron, 

 or to a n-edral (q — \)-acro7i. 



212 



