217 



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 

 parts of the 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. 



XVII. "On Autopolar Polyedra." By the Rev. THOMAS P. 

 KIRKMAN, M.A. Communicated by ARTHUR CAYLEY, Esq., 

 F.R.S. Received June 19, 1856. 



(Abstract.) 



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 

 /3-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 gamics. 



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

 step towards the determination of the number of jp-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. 



