382 Mr. C. W. Merrifield on the Hexahedron 



£(8,3,2,1) = !. 2. 3. 4. 5. 6 + 1.2. 3. 5. 6. 7 + 1.2.3. 5.6. 8 



+ 1.2.3.6.7.8 + 2.3.4.5.6.7 + 2.3.4.6.7.8 



+ 3.4.5.6.7.8. 



And as a corollary, since it may easily be seen that <f>(n, r, fx, e) 

 is always divisible by n when n is a prime and fjur + e < n, it fol- 

 lows that the sum of all the possible products of (any given num- 

 ber) i distinct groups of a given number r of consecutive terms of 

 the series 1, 2, 3, . . . (n — 1) will be divisible by n when n is a 

 prime and ir < n — 1 *. When r = 1, this theorem becomes iden- 

 tical with Wilson's, already referred to. 



Finally, it may be noticed that the number of substitutions of 

 n letters formed by any number of cycles, all of an odd order, 



will be the coefficient of x n in (- ) ,i. e. (\ .3.5...(n — l)) 2 



(the same as the number that can be formed with cycles all of an 

 even order) when n is even, and (\ . 3 . 5 . . . (n— 2) Yn when n 

 is odd f. 



XLIX. Notes on the Hexahedron inscribed in a Sphere, 

 By Charles W. Merrifield, Esq.% 



1. TTS six planes pass, in general, four by four through three 

 X points. 



There is exception, as a singular case, where the intersections 

 of two of the three pairs of opposite planes are parallel. In this 

 case the intersection of the third pair is perpendicular to the 

 parallels, and the inclination of the two planes equal, but opposite. 



Let us consider the intersection of the sphere with any pair 



* For instance, making n=7, r=2, i = 2, 

 1.2.3.4 + 1.2.4.5 + 1.2.5.6 + 2.3.4.5+2.3.5.6+3.4.5.6=784 

 and is divisible by 7. 



f By taking ju=2 in the general theorem, it is an easy inference that if 

 we write 



1 ^ (r+ l)(r+2) ^ (r+ l)(r+2)(r+3)(r+4) + ' 



A. 2t - will be the sum of all the products of 2i integers comprised between 1 

 and r+2i— 1 that can be formed with combinations of i distinct pairs of 

 consecutive integers; thus (e. g.) the coefficient of x 2m in (tan- 1 *) 2 ought 



to be 



m\ ^3 5 2m — 1/ 



which may be easily verified. 

 X Communicated by the Author. 



