Given sets of balls of different colours, in how many ways may they be arranged in line so that no two balls of the same colour shall come together.

If we have two colours only, and the same number ‘m of each colour, there are evidently two arrangements possible; if we have m, m – 1 respectively, only one arrangement is possible; if we have m, m – 2 ; in, m – 3, &c., no arrangement is possible. We may write these results

If the functions be written as in the above figure, then

1. Of any three functions written consecutively on a circular arc or on a diameter, the mean is the product of the extremes. Thus sinA = tanAcosA, 1 = sinAcosecA, and so on.

2. In each triangle the square of the function written at the vertex turned downwards is equal to the sum of the squares of the functions written at the other two vertices. Thus I2 = sin2A + cos2A, and so on.

My object in the following notes is to call attention to some points in integration by successive reduction which may be of use in directing the choice of the particular form for the reduced integral in any given case. The remarks apply chiefly to the binomial differential x(α + bxn)pdx which has been discussed from a different point of view by Dr Muir in vol. iii., p. 100 of the Proceedings.

Étant donnés une surface du second ordre S et deux points A et B, on mène par le point B une sécante qui rencontre la surface aux points C, C′, et le plan polaire du point A au point D. Soient M et M′ les points of la droite AD rencontre les plans qui touchent la surface aux points C et C′. La sécante BD tournant autour du point B, on demands

1°. Le lieu décrit par les points M et M′.

2°. Ce lieu se compose de deux surfaces du second ordre, dont l' une est inépendante de la position du point B, et l' autre Σ dépend de la position de ce point. Chercher ce que devient la surface Σ quand, dans la construction qui donne les points de cette surface, on fait jouer au point A le rôle du point B, et inversement.

3°. Le point A restant fixe, déterminer les positions occupées par le point B quand la surface Σ n' α pas un centre unique à distance finie.