46 Rev. Robert Harley. Prof. Motet's Invariants [Dec. 18, 



p. 215) I ventured to write to him, calling his attention to the theory 

 of criticoids, and suggesting that it might be convenient to compare 

 his results with those obtained by Sir James Cockle, to whom the 

 theory of criticoids is due. 



2. In a footnote to his paper, which I have only lately seen, Pro- 

 fessor Malet says that having consulted the memoirs to which I 

 referred him, he thinks " little similarity will be found between Sir 

 James Cockle's results " and his own. The object of this communica- 

 tion is to show that there is not only similarity, but absolute identity, 

 the two classes of functions considered by Professor Malet coinciding 

 in every point with the ordinary and differential criticoids discussed 

 by Sir James Cockle. The researches of the latter mathematician on 

 this subject have appeared at intervals of time in papers scattered and 

 often fragmentary, in different journals during the last twenty years 

 and more. I propose here to reproduce and present in a connected 

 form the principal portions of those researches ; and in order to 

 facilitate the comparison of results, I will generally adopt the notation 

 employed by Professor Malet. 



3. In a paper, the first of a series,* bearing date 24th December, 

 1861, Sir James Cockle shows that to every form of 



where P 1 , P 3 are functions of x only, there is a cognate form deducible 

 by the following method. Let f(x) or, more simply, f, be any function 

 of x, and substitute / Y for y in the given equation, then 



g + 2Ql f + Q 2 Y=0 ) 



and 



so that g+2P ] ^+(P 3 -Q 2 )/-A{g + (P 1 _Q 1 )/} =0; 



or, developing and reducing by the. substitution of (Q} — P x )/ for 



df 



-f-, we have 

 dx 



* " On Linear Differential Equations of the Second Order," " The Oxford, 

 Cambridge, and Dublin Messenger of Mathematics," vol. i, pp. 118-124, 164-173, 

 241-247. See also " On the Integration of Differential Equations," " Mathe- 

 matical Reprint of the Educational Times," 1868, vol. ix, pp. 105-112. 



