522 The Rev. R. Carmichael on the Singular Solutions 



ordinary electro-magnetic action, either of the fixed conductors 

 upon the moveable one, or of the magnetism of the earth upon 

 it; because in whichever direction the electric current may be 

 passing, either from D to E, or vice versa, the direction of the 

 motion is not affected ; the ball will also, with either direction of 

 current, revolve equally well in opposite directions. The cause 

 of the motion appears to be an intermittent thermic action taking 

 place at the surfaces of contact, at a point a minute distance 

 behind the line of the centre of gravity of the rolling metal. 



These experiments had their origin in a phsenomenon observed 

 by Mr. Fearn of Birmingham, in his electro-gilding establish- 

 ment; — that a tube of brass, | an inch in diameter and 4 feet 

 long, placed upon two horizontal and parallel brass tubes, 1 inch 

 in diameter and 9 feet long, and at right angles to them, and 

 the latter connected with a strong voltaic battery consisting of 

 from 2 to 20 pairs of large zinc and carbon elements, the trans- 

 verse tube immediately began to vibi'ate and finally to roll upon 

 the others. 



Birmingham. 



LXVI. On the Singular Solutions of Differential Equations. By 

 the Rev. Robert Carmichael, M.A., Fellow of Trinity Col- 

 lege, Dublin*. 



THE objects proposed in the following memoir are briefly : — 

 (1) The transformation into a symmetrical form of Clairaut's 

 well-known theorem for the integration of a certain class of ordi- 

 nary differential equations in a single independent variable, and 

 the simultaneous determination of their singular solutions ; the 

 generalization of this transformed type, and the application of 

 the method thence suggested to the determination of the sin- 

 gular solutions of an extensive class of partial differential equa- 

 tions. Of this application copious illustrations are given. 



(2) The examination of the theory attributed to Laplace, for 

 the determination of the singular solutions of differential equa- 

 tions, where they admit of such, from the differential equations 

 themselves, without the knowledge of the general integrals. 



(3) The indication of certain desiderata for the completion of 

 the general theory. 



1. In the memoir by Clairaut, to which allusion has been 

 already made^ a theorem, now well known, was given for the in- 

 tegration of a certain class of ordinary differential equations in a 

 single independent variable, and the determination at the same 

 time of their singular solutions. The difficulty of apprehending 



* Communicated by the Author. 



