456 Rev. R. Harley on the General Quartine, 



is about ^ 5 of the length of the bar, there is not much increase 

 of moment due to the increase of the thickness of the bar; 

 that is, the induced magnetic moment will be practically- 

 independent of the mass of the iron. This statement, para- 

 doxical as it sounds, is not much to be wondered at if we 

 consider that, when the bar becomes very thick, the substance 

 of the bar itself will be forming a kind of internal armature to 

 the free ends of the bar. Thus it seems likely, where thick 

 bar magnets are used in practice, that there may be found 

 waste of material of iron, although in many cases different 

 shapes and the presence of external armatures will modify 

 the condition from the case of the experiments described. 



Glasgow University, 

 October 10, 1888. 



L. On the General Quartine, or the Incriticoid of the Fourth 

 Degree. By the Rev. Robert Harley, M.A., F.R.S* 



CRITICOIDS are those functions of the coefficients of a 

 linear differential equation which remain unaltered when 

 the differential equation is transformed by a change of one of 

 the variables, being analogous in this respect to the critical 

 functions or seminvariants of common algebra. We may 

 divide them into two classes, according as the changed vari- 

 able is the dependent or independent variable. Sir James 

 Cockle, to whom we owe the discovery of these forms f, calls 

 the first class "ordinary," and the second "differential," but 

 in fact both are differential, because both contain differential 

 coefficients. Professor Malet J describes them as Invariants 

 of the first and second class. The functions, however, are 

 not strictly invariants, and the distinction between first and 

 second class hardly seems marked enough. I propose to give 

 the name Decriticoids to those forms which are unaffected by 

 a change of the dependent variable, and the name Incriticoids 

 to those which are unaffected by a change of the independent 

 variable. A decriticoid of the mih degree may be called an 

 m-ide, and an incriticoid of the same degree an m-ine. 



* Communicated by the Author. 



t Harley. " Professor Malet's Classes of Invariants identified with Sir 

 James Cockle's Criticoids." Proceedings of the Koyal Society for 1884, 

 vol. xxxviii. pp. 45-57. 



% Malet. " On a Class of Invariants," Philosophical Transactions for 

 1882, Part III. pp. 751-776. 



