this type so that m, is as great as possible; and let J, be the Abelian subgroup of I, of order m2. Then J, has no operation other than identity in common with J, or with any subgroup conjugate to J1; also no two subgroups conjugate to J2 have a common operation other than identity, and m, and N/m2 are relatively prime. All these statements may be proved exactly as in the former case.

3

2

If the subgroups of G of order 2μ (μ odd) are still not exhausted, a subgroup I, of order 2m, containing an Abelian subgroup J, of order m, may be chosen in the same way as before; and the process may be continued till all subgroups of G of the type in question. are exhausted. Now J, is one of N/2m, conjugate subgroups and each contains m1 – 1 operations which enter into no other subgroup conjugate to J, or to J, or J,.... the subgroups conjugate to J1, J2, J., contain

N

2mi

...

3

2m3

Hence

distinct operations other than identity. If I actually existed, this number would be equal to or greater than N, which is impossible. Hence there can at most be only two sets of conjugate subgroups such as I, and I.

It was shewn in section 1 that each of the m-1 operations of J, other than identity can be represented in m, distinct ways as the product of two operations of order two. Similarly each of the m2-1 operations other than identity of J2, if it exists, can be represented as the product of two operations of order two in my distinct ways. Moreover these and the operations conjugate to them are the only ones which can be represented as the product of two non-permutable operations of order two. Now G contains

(2n+1)(2" - 1)

operations of order two, and any one of these is permutable with exactly 2"-1. Hence the number of products of the form AB, where A and B are non-permutable operations of order two and the sequence is essential, is

(2nk + 1)(2" - 1) 2′′k (2" − 1) = Nk (2n − 1).

On the other hand as shewn above this number is

These conditions are obviously inconsistent. Hence I, does exist, and

2

m1 + m2 = 2 {k (2′′ − 1) + 1}.

It follows that, m, and m2 being positive numbers of which m, is the greater,

m1 > 2"+1-k.

On the other hand, since no two operations of order two contained in I, are permutable, while G contains only 2"k+1 subgroups of order 2",

and à fortiori since is less than k, and 2"k+1 is positive,

The group G can therefore only exist if k is unity, and this necessarily involves that

l is zero. Hence

N = (2" + 1) 2′′ (2" - 1), m1 = 2′′ + 1, m2 = 2" — 1,

and these are the only values of N, m, and m, consistent with the existence of a group G having the required property.

Since G is simple, it can be represented as a substitution group of degree 2+1. The subgroup of degree 2", which leaves one symbol unchanged, has a self-conjugate Abelian subgroup of order 2", and 2" conjugate Abelian subgroups of order 2" -1; the latter having no common substitutions except identity.

Hence the subgroup of G which leaves one symbol unchanged is doubly-transitive in the remaining 2" symbols; and therefore G can be represented as a triply-transitive group of degree 2" +1.

The Abelian subgroup of order 2" 1 which transforms a subgroup of degree 2" is shewn in an appended note to be cyclical. Assuming for the present this result, the subgroups of G of order 2" (2"-1) are doubly-transitive groups of known type.

Now G contains just 2"-1 operations of order two which transform each operation of a cyclical subgroup of degree 2" 1 into its inverse. Since each of these leaves only one symbol unchanged, each must interchange the two symbols left unaltered

by the cyclical subgroup of order 2" - 1. But there are only just exactly 2"-1 substitutions of order two in the 2"+1 symbols which satisfy these conditions. Hence for a given value of n the group, if it exists, is unique. That such groups exist congruences

for all values

of n

values of n is known*. In fact the system of

actually define such a group; and the permutations of the 2" + 1 symbols

which are effected by the above system of congruences, actually represent it as a triplytransitive group of degree 2"+ 1.

The set of groups thus arrived at are the analogues of group (iii) above.

Finally, every group of even order, which does not belong to one of the three sets thus determined, must contain operations of even order other than operations of order

two.

NOTE.

Let H be an Abelian group of order 2" whose operations, except identity, are all of order two; and suppose if possible that H admits two permutable isomorphisms of prime order p one of which is not a power of the other, such that no operation of order two is left unchanged by any isomorphism generated by the two. So far as a set of p2 operations of H are concerned the two isomorphisms, being permutable, must have the form

* Moore: "On a doubly-infinite series of simple groups," Chicago Congress Papers (1893); Burnside: "On a class of groups defined by congruences," Proc. L. M. S. Vol. xxv. (1894).

being the p2 operations. Moreover any cycle of an isomorphism generated by these two has the form

(Ar, 8 Ar+x,

8+ y, ......

Ar+(p−1) x, 8+ (p−1) y),

the suffixes being reduced mod. p.

Since no operation of H except identity is left unchanged by any one of these isomorphisms, the product of the p operations in any one of the cycles must give the identical operation.

operations of H except It is therefore cyclical*. subgroup of order p

The supposition made therefore leads to a contradiction. Hence if H admits a group of isomorphisms of order pm, no one of which leaves any identity unchanged, this group has only a single subgroup of order p. If then pm is the highest power of p which divides 2"-1, the in the Abelian group of order 2" - 1, considered above, is cyclical. Hence the Abelian group is itself cyclical.

XII.

On Green's Function for a Circular Disc, with applications to Electrostatic Problems. By E. W. HOBSON, SC.D., F.R.S.

[Received 7 October 1899.]

THE main object of the present communication is to obtain the Green's function for the circular disc, and for the spherical bowl. The function for these cases does not appear to have been given before in an explicit form, although expressions for the electric density on a conducting disc or bowl under the action of an influencing point have been obtained by Lord Kelvin by means of a series of inversions. The method employed is the powerful one devised by Sommerfeld and explained fully by him in the paper referred to below. The application of this method given in the present paper may serve as an example of the simplicity which the consideration of multiple spaces introduces into the treatment of some potential problems which have hitherto only been. attacked by indirect and more ponderous methods.

THE SYSTEM OF PERI-POLAR COORDINATES.

1. The system of coordinates which we shall use is that known as peri-polar coordinates, and was introduced by C. Neumann* for the problem of electric distribution in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate system; in order to measure the position of any point P, let a plane PAB be drawn through P containing the axis of the circle and intersecting the circumference of the PA PB

circle in A and B; the coordinates of P are then taken to be plog @ which is the angle APB, and the angle made by the plane APB with a fixed plane through the axis of the circle. In order that all points in space may be represented uniquely by this system, we agree that shall be restricted to have values between -T and π, a discontinuity in the value of 0 arising as we pass through the circle, so that at points within the circumference of the circle, is equal to T, on the upper side of the circle, and to - on the lower side of the circle, the value of being zero at all points in the plane of the circle which are outside its circumference.

* Theorie der Elektricitäts- und Wärme-Vertheilung in einem Ringe. Halle, 1864.

As