TRANSACTIONS OF SECTION A. 745 
7. On some Arithmetical Functions connected with the Elliptic Functions 
of 3 K. By Dr. J. W. L. Gratsner, F.R.S. 
8. On Systems of Simultaneous Linear Differential Equations. 
By A. R. Forsyru, F.R.S. 
9. Chess Problem.’ By Lieut.-Col. Attan Cunnincuam, R.F. 
‘To find the number of different positions after two moves on each side at the 
game of chess.’ 
This is—in a mathematical sense—a fairly simple problem in combinations ; but 
the rules of chess introduce into it such a number of variations requiring separate 
estimation, as to make the complete solution a pretty laborious task. Without 
» great care in the detail there is much risk of omission, also of counting the same 
position twice, and of counting positions which cannot be formed in actual play. 
On account of the great historic interest of the game of chess, it is thought 
worth while to publish the results. The following is an abstract :— 
I. Pawns only moving . ¢ . ‘ ; ‘ . 16,556 
4 II. Captures by pawns; at least one piece moved. : 347 
: Ill. No captures by pawns; both sides move at least one 
q piece : : : : : : : : . 19,441 
1 IV. No captures by pawns; one side moves pawns only, 
the other side moves at least one piece. : . 35,438 
Grand total =. 21,782 
10. On a Remarkable Circle through two Points of a Conic. 
By Professor Geneszr, M.A. 
A, B are two fixed points of a conic, C the pole of AB, P a variable point os 
the curve. Through C an antiparallel is drawn to ABwith respect to the angle 
APB, meeting its arms in Q, Q’; in other words, QCQ’ is drawn so that the points 
A, B, Q’,Q lie on a circle. This circle is invariable. 
__ AQ’, BQ meet on the conic, at P’, say ; then PP’ passes through a fixed point 
T (the pole of AB with respect to the circle). 
Thus, using C, the circle can, by means of the ruler, be transformed into the 
conic, or, using T, the conic can be retransformed into the circle. 
It will be noticed that the point T has the property that for any chord PP’ 
through it the sum of the angles APB, AP’B with a proper convention is constant. 
11. Ferrel’s Theory of the Winds. By Cuartes Cuampgrs, F.R.S. 
The object of this paper is to point out a defect in Dr. Ferrel’s analytical in- 
vestigation of the motions of the atmosphere, to supply that defect, and to substi- 
tute legitimate interpretation and geometrical illustrations of the analytical results 
arrived at for a misleading and irrelevant exposition given in several of the revisions 
% of Dr. Ferrel’s research that have been published from time to time during the last 
thirty years. 
, 
e 
q DxrpaRtMEentT IJ.—GeEnERAL Puysics anD Enncrrorysis. 
1. On a Method of determining in Absolute Measure the Magnetic Suscepti- 
bility of Diamagnetic and Feebly Magnetic Solids. By Sir Witviam 
a Tuomson, D.C.L., DL.D., F.R.S. 
The communication was suggested from two directions in which the subject 
__ had been treated—(1) Professor Riicker’s investigations of the magnetic suscepti- 
‘ This problem has been published iz extenso in the Royal Engineers’ Journal for 1889. 
— 1890. 3 
