TRANSACTIONS OF THE SECTIONS. 21 



(II.) The difterentlal equation 



y"'+2xy'—y=0, or l-'[-2x!)f—yx"^=0, 

 has for its general integral 



Putting, for simplicity, 



y+x-=:z% 3ya:+2a;'+c=2«', 

 we find 



_dy __ 2(3yg4-2.r^ + c) _ 1 



do QxQiyz+2x'+c)—\2{y+x''f \x-z) ' 



dx _ 2(3y«+2rH-c) 



dc Q{y-\-2x')(;iyx+2x^-\-c)~'2,4:x{y+x')'- 3(a:— 2)=*' 

 and consequently 



a:'=f =. J- (8) 



dy x—z 



The systems (5) and (6) lead to the relation ?/+a;^=0, which is not a singular solu- 

 tion, as it also belongs to the case of exception (7). But the systems (3) and (4) 

 lead to 



z—x=(X>, or »/y+x^—x=aci , (") 



a relation which is in a certain sense a singular solution of the given equation. 

 To show this, put 



»/y+x^—xs=A, or y=2Ax+A?, 

 and we deduce 



2A 



On the other hand, the curves represented by the general integral have, after (8), 

 for the coefficient of the direction of their tangent 



x'=-L 

 A 



If A increases indefinitely, these two values of x' tend towards the common limit 

 zero. In a certain sense, therefore, the equation (9) represents a singular solution 

 of the given equation. 



These two examples are sufficient to explain the apparent exceptions to rule II., 

 which we may consequently regard as giving all the singular solutions which are 

 not at the same time particular integrals. 



On Singular Solutions. By Professor Henry J. Stephen Smith, F.R.S. 



On the Effect of Quadric Transformation on the Sinc/ular Points of a Curve. 

 By Professor Henky J. Stephen Smith, F.B.S, 



Note on Continued Fractions. By Professor Heney J. Stephen Smith, F.R.S. 



Contributions to the Mathematics of the Chessboard. By H. Martyn Taylor, 

 M.A., Felloiv and Tutor of Trinity College, Cambridge. 



The object of the paper was to ascertain the relative values of the pieces on a 

 chessboard. If a piece be placed on a square of a chessboard, the number of squares 

 it commands depends in general on its position. If we calculate the average num- 



